Although steam navigation and technology were developing rapidly, the 19th century was in a way the golden age of sail; for sailing ships and sailing technology were developing rapidly also. What now seems very characteristic of sail may be things that were only really developed in the 19th century. For instance, from Napoleonic warships to the Clipper ships of the China tea trade, we tend to think of sailing ships as three-masted, full square-rigged ships, with some exceptions, like two-masted brigs. All of the rates of ship in the Royal Navy, from First Rate Ship-of-the Line down to unrated Sloop-of-War, were three-masted, full-rigged ships.
However, later 19th century ships became larger, with four, five, or more masts, and with the substitution of fore-and-aft sails for cross-yards and square sails. Thus, the types developed of a bark, with only fore-and-aft sails on the final mast, the barkentine, with only square-rigged sails on the first mast, and the brigantine, a brig with square-rigged sails only on the foremast. Barks and barkentines would grow to four or five masts.
Before the Napoleonic era, the size of ships was limited by a flaw in the construction technique. Longer ships tended to sag at the ends after a while, because the narrower beam supported the weight of the ship less well. When it was realized that you could cross-brace the ribs, and make the hulls more rigid, ships could become larger and longer.
The real revolution, however, was the introduction of iron and steel. A steel ship is going to be a lot stronger and stiffer than whatever you do to a wooden ship. Also, with steel masts and yards and steel wire in the rigging, a ship could take a lot more punishment from the wind than wood and rope could ever endure. Sailing ships became windjammers, and things that used to be terrors for sailors, like the passage around Cape Horn, became routine. 
A full gale just meant greater speed, not the danger of rigging carrying away and the ship becoming disabled. Although it was realized that a full fore-and-aft rigged ship could sail closer to the wind, the square-rigged ships were still sturdier and easier to handle in heavy weather. Schooners, which did grow to up to seven masts, therefore tended to be more common in coastal navigation and might avoid the high seas.
The ultimate expression of steel sailing may have been the German ship Preußen, the only five-masted full square-rigged ship built. 
This was also not even a ship of the 19th century, but of the 20th. The Preußen took advantage of all the virtues of a sailing ship. Over routes with reliable winds, the ship carried on trade from Germany to Chile, bringing back bulk nitrates. Its top speed was 20 knots, which was faster than the Pre-Dreadnought battleships of the era. When the wind is free, you've got a pretty good deal going. Unfortunately, its success ended up being its downfall. In 1910, cruising outbound down the English Channel, the Preußen was cut off by a ferry, whose pilot did not believe that a sailing ship could be making 16 knots. Damaged enough to require a tow, the weather prevented a tug from arriving before the ship went on the rocks and was broken. It would be pretty much the last of its kind.
After World War I, the world changed. The ultimate drawback of sailing ships was that they were complicated and labor intensive. You had to "learn the ropes," and so the ships needed crews of experienced sailors. But the world where sailors did not even need a passport, and might end up only dimly aware of where they were even from, disappeared. Where fuel might be expensive and hard to find overseas in the 19th century, cheap and plentiful oil was now the norm. With someone to fix the engines, a merchant ship did not otherwise need much of a crew. So although the windjammers kept up and were economical until World War I, they suddenly were gone, taking much of the traditional culture of sailors with them. The height of technology that they had achieved would only be preserved for naval training, show, and sport. Now, in the romance of sail, it is rare to see 19th century steel ships. Movies stick to the 18th century, the Napoleonic Era, or earlier; and one really might never know from Hollywood that sailing ships with more than three masts and iron hulls ever existed. Even "Old Ironsides," the U.S.S. Constitution, is, after all, a wooden ship.
Ancient sailing ships might only carry one sail. When a second sail was added, this became, logically, the "topsail." Eventually, we got a third sail, a "topgallant," and then a fourth, the "royal." 
These would be specified, of course, by mast. The "mainsail" is on the mainmast, while the "fore topgallant sail" would be on the foremast. The Clipper ships added a fifth sail, the "skysail," and sometimes even a sixth, the "moonsail."
However, this trend was soon overtaken by another development. Previously, sails had lines woven into them, such that the sail could be "reefed," i.e. part of it tied to the yard so that less area was exposed to the wind. This was done in heavier weather, and of course required that crewmen get out on the yard, in such weather, to do the tying. To say the least, this was dangerous work. So eventually, it was reckoned that it was easier to have more yards and narrower sails. First the topsails and then the topgallants were doubled and reduced. In port, the difference between the upper and the lower sails was conspicuous, since the yard of the upper sails was designed to be hoisted off the lower, which meant that when the sails were in, the upper yard lay immediately above the lower yards. Thus, a ship with six sails on a mast, like the Preußen, only went up to a royal, while an older ship might have had a skysail or a moonsail to have as many on the mast. Where the upper topgallant will be hoisted, this means that the royal yard will be hoisted also.
With a "middlemast" (German Mittelmast), between the main and the mizzen, the masts of the Preußen went back to a "jiggermast" (Kreuzmast). 
Schooners, which often had several more masts, tended to avoid using "middlemast" and added in sequence "diver," "pusher," and "spanker" masts. Or some simply numbered extra masts. The names of the seven masts of the schooner Thomas W. Lawson (1902-1907) kept being changed before, during, and after construction -- the names in the image are attested by a letter from the captain. The Lawson does not seem to have been a very successful ship and was finally lost on the rocks of the Scilly Isles, joining Admiral Shovell. 
If we gave Western names to all the nine masts of the great Ming Dynasty baochuan, or treasure ships, we would need all these expedients of naming.
Even square-rigged ships end up carrying a good bit of fore-and-aft-rigged sails. In terms of numbers, these may mostly be "staysails," i.e. sails rigged on the "stays," which are already part of the standing rigging of a ship. They hold the masts down towards the bow of the ship. The lines that hold the masts toward the stern are the "backstays," while those that hold the masts to each side are the "shrouds." The shrouds are usually woven into rope-ladders, so that the crew can easily climb them up to the yards. 
There are generally more backstays than stays, since they must bear a greater load as the wind fills the sails, and they are tied down in pairs closer to the masts on the sides of the ship. The stays are therefore better situated to hold a sail, being on the centerline, and generally traversing the distance between the masts, or from the Foremast to the bowsprint. A stay that goes down onto the body of the bowsprint will carry a "jib" rather than a staysail. There can be several jibs. Finally, the last mast astern even on a fully square-rigged ship carries a large fore-and-aft sail, set on a "boom" below and a "gaff" above. The sail will be the "spanker," but this name may vary depending on the number of masts and type of rig. Names of masts like "jigger," "spanker," and "driver," mainly on schooners, are derived from or affixed to the corresponding fore-and-aft sail. Above the spanker may be a "gaff topsail," which can be doubled like other topsails. All these fore-and-aft sails, except the spanker, will tend to be triangular in shape, unlike the rectangular sails on the cross-yards.
Standing on the deck of a ship, which can be pointed towards any direction on the horizon, it helps to have an unambiguous way to refer to direction. 
This is done relative to the structure of the ship. Sailors are infamous for avoiding landlubber terminology like "left" and "right," which, being relative to one's own body, are ambiguous if spoken to persons who may be facing in different directions than the speaker. "Starboard" neatly sums up the periphrasis, "on the right side of the ship, if one is facing towards the bow." Since "starboard" and "port" traverse a great deal of arc, they are traditionally divided in three, with the "beam" on each side exactly 90o from the centerline of the ship, the "bow" in the quarter between the beam and the bow, and the "quarter" in the quarter between the beam and the stern. The direction towards the actual bow will be "ahead" or "dead ahead," while towards the actual stern, "astern" or "dead astern." This divides direction into eight points, with the image showing the sixteen point compass revolving around it. North by Northwest (NNW) is the name of a 1959 Alfred Hitchcock movie, whose action culminates on Mt. Rushmore. (In some systems, stop animation with ESC key.)
The "quarters" may get their name from the "quarterdeck," which is the area between the last mast astern and the wheel, where the officers of the ship tend to stand, observing both the condition of the ship and the direction of its motion, and giving instructions to the crew and the helmsman. 
The quarterdeck may also actually be a deck higher than the waist of the ship, and often lower than the last deck at the stern, the poopdeck. Sometimes the ship's helm is under the poopdeck. Or, as became more common in the history of sailing ship construction, the whole deck may be "flush," i.e. a single level, which certainly would make it easier to run back and forth between the different masts. On modern warships, a certain area on the ship may be designated the "quarterdeck" for ceremonial purposes. This does not need to be anywhere near where the quarterdeck would have been on a sailing ship.
After the Napoleonic Wars, newer ships tended to be flush deck (i.e. same level the length of the ship) and the bulwarks along the sides of warships came to be built up, in order to protect crew on the deck from gunfire. However, this began to obscure the view of officers on the quarterdeck.
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| Stair to port bridge on Cruiser Olympia, pilot house under tarp. |
The Officer of the Watch, the Deck and the Conn
Philosophy of History, Military History