The concepts of substance and essence are among the most fundamental in metaphysics. They are also among the most sharply questioned, in both Eastern and Western philosophy. Today, "essentialism," the belief in essences, is regarded a fallacy in much academic opinion, both sensible and foolish. Nevertheless, what the ideas represent is something that it is difficult to do without, in both ordinary language and any serious ontology.
Some simple definitions are in order. A "substance" has certain characteristics. It is durable, separable, and identical. An "essence" is that which makes something what it is. The definitions of substance and essence may both be said to express what it is that makes them what they are, i.e. their essences, if the essences are themselves durable, etc.
A substance as "durable" means that it persists over time. It endures. It may come into existence, or cease to exist (as in Aristotle), or it may be uncreated or indestructible (as in Plato, Descartes, Spinoza, or Leibniz), but either way, it has an extended existence in time. A substance as "separable" means that it's existence is not dependent on other things. It exists independently, and it can be separated from other things that exist. A substance as "identical" means that it has an identity, in which it is the same thing as itself, or in which it has an identity as the member of a certain kind -- the same as it endures over time, or as it is separated from other things.
The essence as what "makes something what it is" implies that the something is something, i.e. a particular kind of thing, i.e. a dog, radio, planet, number, etc. But substance is not necessarily about kinds of things, since what is durable, separable, and identical may be an individual where, as such, what it is is irrelevant. Thus, Aristotle distinguished "primary substances," i.e. individuals, from "secondary substances," i.e. kinds. A kind of thing is then to be associated with an essence. The metaphysics of the essences of kinds get us into the Problem of Universals. Whether individuals have an essence is a good question. The question gets us into the issue of Naming. Using Frege's distinction between sense and reference, it looks like what makes a particular individual that individual is not in the sense, which can always specify more than one individual, but in the reference. Although it is common to assume, as did Frege, that sense determines reference, this generates paradoxes and has been ably refuted by Jerrold Katz.
Aristotle's terminology in these matters now looks a little confusing. The Greek word for "substance" was ousía, from oûsa, the feminine participle of "to be" (infinite, eînai). Thus, the word looks more like Latin essentia, "essence," which is from the infinite of "to be," esse. Terminologically, Aristotle does not seem to have clearly distinguished between substance and essence. On the other hand, as substantia in Latin appears to mean "stand" (stare) "under" (sub), there is a word corresponding in meaning in Greek: hypokeímenon (as a neuter passive participle), "lie" (keîmai) "under" (hypó). Aristotle uses this to mean "matter" in his sense, which is not substance, precisely because it is not separable. Aristotelian matter is merely potential and is parasitic on "form," which is the actuality (enérgeia) of the thing. Potentiality does not have actual existence and so is not separable. On the other hand, Aristotle's matter is what allows him to avoid substances that are uncreated and indestructible, since the matter "underlies" the transformation of one substance into another.
In Greek philosophy, on top of substance and essence, we get the issue of the ontôs ónta, the "beingly beings," i.e. what things most truly exist. For Plato, that would be the essences of kinds, the Forms (an eîdos or idéa), in the World of Being; for Aristotle, it is the actuality of the individual, in the form again (eîdos or, in Latin, species); and for Descartes, just so we move across the board, it is, for natural objects, in the matter, which is essentially extension. Although the Cartesian view of matter now seems the most natural and obvious meaning of "substance," it nevertheless is the conception that has suffered the most from developments in science. While Descartes believed, as many still do, that matter is a solid plenum of stuff, in physics matter has disintegrated into a blizzard of abstract features in largely, or entirely, empty space. Since Einstein made mass equivalent to energy, we might say, to the delight of Aristotle, that matter has disintegrated into enérgeia.
Substantia becomes the term for ousía in Latin, perhaps with the sense that what endures may undergo superficial changes and so "underlies" such apparent changes. Such changes, indeed, are what we still say are not "substantial." In terms of essence, superficial changes do not change what the thing is, and so the changes are merely "accidential" rather than "essential." While the contrasts between "substance and attribute" and "essence and accident" are now standard, the original combination of substance and essence in Greek ousía we see in the occasional use of the expression "substance and accident." To fully untangle them, we need to be clear that essence is defined by attributes. What makes the thing what it is are certain characteristics, and these inhere in the durable and separable substance. Indeed, they identify it, as a member of its kind.
A very ancient rejection of substance and essence altogether, or their equivalents in Indian philosophy, began in Buddhism. Since the approach of Buddhism to the world is to break attachments, so that one does not suffer because of relationships to things, a simple way to do that is to say, in effect, that there are no things. If nothing is substantial or has any essence, this will do that job. What we get instead are the doctrines of "momentariness," "no self nature," and "relative existence." If everything exists only momentarily, then nothing is durable, and we lose that characteristic of substances. If there is no self nature, then there is nothing in things that makes them what they are, and we lose the existence of essences. If things only exist relative to other things, then (1) nothing exists independently and we lose that characteristic of substance, and (2) nothing has its own character, so we lose that characteristic of essence. So what is actually there? Well, what we see is the "form" of things, the external appearance. Since there is no self nature and things only have relative existence, what are things in themselves when we take away everything else? Well, Emptiness. This is not nothingness (a major heresy), but neither existence nor non-existence nor both nor neither. In other words, we can't say or comprehend what is there. Later, in Mahâyâna Budddhism, we get the doctrine of the Heart Sutra that "Emptiness is Form, and Form is Emptiness."
Unfortunately, Buddhism always had difficulty with the implications of all this fundamental metaphysics. The loss of substance and essence takes with it identity, so that it becomes difficult to say that an individual, like the Buddha, achieves Enlightnment and Salvation. The individual, in fact, does not survive beyond the moment, and so it is a different being who achieves Enlightenment from the one who existed previously, and a different being in turn who achieves Salvation. Buddhism attempts to substitute causality for substance, so that what I am now is simply caused by what I was before. Unfortunately, this does not restore identity. If I make a tuna sandwich, and so cause its existence, this does not mean I am the tuna sandwich. Causal connections can be within substances or pass between them, and the identity relation is contributed by the substance, not by the causality. In the end, Buddhism seems to settle into the notion of "provisonal existence," which is durable and identical, and then, with some other expedients, ceases to worry about the matter. The popular belief, indeed, is that Buddhism is about finding one's true self, not about finding that there is no self at all (anatma or anatta, "No Self").
In Mediaeval Europe we get the rejection of the reality of essence by the Nominalists. To them, universals are just "names" (nomina). Their very sensible motivation would seem to be that what we see and experience in the world are concrete individuals, not abstract universals or essences. Unfortunately, the difficulty they would always have is that individuals do not become members of kinds just because we happen to apply a particular name to them, but we apply a particular name to individuals because we recognize a feature in them that matches up with the meaning of the word. The Nominalists thus not only rejected the reality of essences, but they tended to overlook the abstract content of meaning as well, focusing only on the tangible things, the object and the word. Both these tendences survived well into Modern Philosophy and down to the present.
Meanwhile, a focus on the tangible had earlier produced a striking assertion. The Islamic philosopher Al-Ghazzâlî (d.1111) had noted that the causal connection itself was an intangible feature that purportedly connected different tangible objects. So he did not believe that it needed to be there. Actually, Ghazzâlî did believe in causality; he just did not believe that it was where we thought it was. Instead, God alone is the cause of everthing that happens -- this is Islamic "Occasionalism" (since an event is not really a cause, but simply the "occasion," for an effect), a term offered by Malbranche, and a doctrine still used by Spinoza. Note what this would do to Buddhist metaphysics, where there is no God: Its last realistic principle would be gone. But this would never happen in Buddhism, where the Buddha himself asserted that there were causes, i.e. that there is a cause for suffering.
In David Hume (d.1776) we have a true Götterdämmerung of metaphysics. All together we get the Nominalist rejection of essences, Ghazzâlî's critique of causality, and the Buddhist rejection of substance. With all of these, Hume focuses on the tangible, with the added Empiricist notion that the only contents of the mind are images, because that is what experience delivers. Thus there are no "abstract ideas," a thesis easily confirmed when abstractions are discovered not to be images, e.g. a "human being" cannot be imagined without attributes that always characterize particular human beings -- i.e. neither short nor tall, neither fat nor thin, neither light nor dark, neither male nor female, etc. Since a substratum to experience is not visible in experience, nor the abstract essence visible in the concrete individual, these are going to fall to Hume's critique -- although, as we have seen, Hume himself initially offers to abandon his theory if a concept can be cited that he cannot trace to an antecedent image. Evidently realizing the difficulty this would create for him, Hume then shifted to arguing that a concept, far from refuting his theory, is meaningless if he cannot trace it to an antecedent image.
Abandoning substance and essence, Hume would be left with the same difficulties as his precedessors. How an object maintains a durable indentity over time, Hume cannot not account for. How he would even know whether a particular word applies or does not apply to a particular object would also be a difficulty. Since the British Empiricists all played billiards, we could ask how Hume identifies a billiard ball (which figures in the discussions of causality by Locke, Berkeley, and Hume). That it is round, indeed a sphere, requires the recognition of an abstract feature. It does not matter that the word is applied to the ball, since we want to know what it is about the ball that would merit the application of the word to it. The word "ball" has no affinity with the ivory (or plastic) object on the pool table.
What Hume must do, which he is quite justified in doing given the rest of his thought, is simply to say that we do apply the word, just as we otherwise use the concepts of causality and of vice and virtue, without being able to rational justify this action. For Hume, as it happens, states quite openly that the Skepticism of his thought "has little or no influence on practice" [Treatise of Human Nature, Shelby-Bigge edition, Oxford, 1888, 1968, p. 469]. The custom and the habit of humanity are Hume's ultimate justification, even when rational justification is exposed as vaccuous.
Hume's discussion of identity is revealing:
...we may observe, that the view of any one object is not sufficient to convey the idea of identity... One single object conveys the idea of unity, not that of identity.On the other hand, a multiplicity of objects can never convey this idea, however resembling they may be suppos'd. The mind always pronounces the one not to be the other, and considers them as forming two, three, or any determinate number of objects, whose existences are entirely distinct and independent.
Since then both number and unity are incompatible with the relation of identity, it must lie in something that is neither of them. But to tell the truth, at first sight this seems utterly impossible. Betwixt unity and number there can be no medium; no more than betwixt existence and non-existence. After one object is suppos'd to exist, we must either suppose another also to exist; in which case we have the idea of number: Or we must suppose it not to exist; in which case the first object remains at unity.
To remove this difficulty, let us have recourse to the idea of time or duration. I have already observ'd, that time in a strict sense, implies succession, and that when we apply its idea to any unchangeable object,'tis only by a fiction of the imagination, by which the unchangeable object is suppos'd to participate of the changes of the co-existent objects, and in particular of that of our perceptions. This fiction of the imagination almost universally takes place; and 'tis by means of it, that a single object, plac'd before us, and survey'd for any time without our discovering in it any interruption or variation, is able to give us a notion of identity... Here then is an idea, which is a medium betwixt unity and number; or more properly speaking, is either of them, according to the view, in which we take it: And this idea we call that of identity. [op.cit. pp. 200-201, boldface added]
Hume must have recourse to many "fictions of the imagination" in order to salvage many of the principles of ordinary belief and experience. Recourse to "time or duration" will not be sufficient in the matter, since with the perception of successive objects it is perfectly conceivable (Hume's own criterion of the possible) that we have successive objects (number) rather than the same object (unity and identity). Thus, even if an object is "survey'd for any time without our discovering in it any interruption or variation," this is not of itself "able to give us a notion of identity" -- any more, I might add, than the regularities of constant conjunction "give us a notion" of causality without, as Hume says, the subjective psychological expectation that is created. The "fiction of the imagination" is that there are some "unchangeable objects" which undergo some, but not all, of the duration that we perceive in experience. The underlying, unchangeable reality is itself invisible. It is not enough that it looks the same, for we know that, as we turn our back on the pool table, our trickster friend can, behind our back, switch the original cue ball for one from another table. Examining the ball, when we may not have paid very close attention to the original one, may not reveal the truth.
For Immanuel Kant (d.1804), what the mind supplies is not a "fiction of the imagination" but a concept, a "category," that, a priori, is necessary for the coherence of the world. He understood that Hume could hardly disagree with this but had had a difficulty accounting for its origin and the necessity of its agreement with experience. The former is easier when we allow that abstract concepts are not images, so that they may be innate without our being aware of them (which Locke could not allow). The latter, and part of the former, depends on Kant's own theory that experience and phenomenal objects themselves are generated by the activity of the mind, reflecting rules without which consciousness would not exist, and which are then sewn into the fabric of the tangible world.
Thus, when we go about our lives and ask, "Is this the soap that was here earlier?" or "What makes this shoe a "pump'?" we presuppose that the earlier soap is a durable object that can be identical to the present one, and that there are features that make a "pump" what it is. To the nature of the world that makes these sensible questions and presuppositions realistic we supply the concepts "substance" and "essence." What is really behind the appearances, what is really enduring, we cannot say. Concepts of substance in Descartes, Spinoza, and Leibniz are of indestructible and eternal subsances: matter, soul, God, and monads. Substance, however, does not necessarily require indestructibility. Aristotle's substances come into being and pass out of being, which is what happens with shoes and other ordinary objects. The basic meaning of "substance" thus does not resolve some of the most important questions about substance, e.g. whether there is a soul, God, etc. Indeed, it doesn't even answer the question whether there is "matter" in the sense meant by Democritus, Descartes, or materialists since them.
At the same time, the meaning of "essence" does not answer basic questions about essence. What makes something what it is may not involve any necessity or causality within the object itself. Thus, human artifacts are meaningful only in relation to human purposes. A shoe is not a natural kind but an object that will decay and disappear from the world unless we use and maintain it. The essence of a shoe only exists as an artifact of human consciousness. At the same time, there are natural kinds, 
and what makes a deer a deer or an igneous rock an igneous rock depends on causation internal to the things. The features of an essence must be held together by something, and it looks like this must be the different modes of necessity, which have been examined elsewhere. Thus, the laws of nature make many things what they are, but logic and a priori metaphysical truths underlie more fundamental things. Truths of value, of justice, goodness, and beauty go beyond these, into modes of purposive truth. At the same time, a definition of something can just be made up and stipulated. Things can even be done this way in mathematics. The necessity of the essence there is that of a matter of fact, or of "perfect" necessity.
In Kantian terms, essences require synthesis. Kant thought of synthesis in terms of propositions, i.e. synthetic propositions, with analytic propositions then merely unpacking in the predicate the meaning already put together in the subject. He does not get into the question of what put the meaning together in the subject in the first place. However, his notion about propositions, that a ground connects subject and predicate, works just as well for the structure of meaning in the subject. A deer is made by its DNA and defined by the zoologist using diagnostic features of its anatomy. Our concept of the deer, then, it put together in ultimate dependence on the causal ground of the DNA. Since that is not something that, presently, can be examined directly, we infer it indirectly. Now, at least, from modern biochemistry we know that the DNA is there and is the inner spring of the process -- something Hume doubted we could ever know. Other essences -- of logic, mathematics, metaphysics -- do not require an examination of physical causes.
A Kantian theory of substance and essence is thus only the beginning of how they work. They do not just leap out at us from the objects. They are, however, indispensible, and themselves are part of the a priori structure of knowledge that metaphysics must account for.