Mr. Gonzales has laid out the basic system. I'll take the liberty of adding a little bit, in the hope of clarifying things slightly.
Prior to the mid-nineteenth century it was customary for all yards to slide up and down the masts as part of the process of setting and furling the sails. The exceptions were the lowest yards on the three masts: the fore lower yard, main lower yard, and crossjack yard. (The latter normally didn't have a sail on it during that period.) Each yard was raised and lowered by means of a tackle rigged to its center. That tackle was called the jeers in the case of the lower yards, and the halyard in the cases of all the others. By the end of the eighteenth century it was also common, especially in warships, for the lower yards, once they'd been hoisted into position by the jeers, to be secured semi-permanently by another, non-adjustable tackle called the truss.
The lifts were separate, lighter pieces of gear secured to the yardarms (i.e., near the tips of the yards. Each yard has two yardarms - one at each end.). The purpose of the lift was to keep the yard horizontal - or, under some circumstances, to tilt it one way or the other. The lower yards could, by means of the lifts and the braces, be used as derricks for handling cargo and the ship's boats. And certain sailing evolutions called for the upper yards to be tilted, so only a triangular portion of the sail would be exposed to the wind.
By the Cutty Sark's day the system had evolved considerably - and, in general, simplified. She, like most other good-sized ships of the period, has double topsails. The double topsail was conceived as a labor-saving device. (Some people have the notion that its purpose was to increase the amount of sail the ship set; in fact it did precisely the opposite. All other things being equal, a ship with double topsails has less sail area than one with single topsails. Most of the big American clipper ships switched from single to double topsails as their careers went on.) The old single topsail was, in effect, cut in half, each half being reasonably convenient for a relatively small group of men to handle. A new yard, the lower topsail yard, was inserted into the picture. It was generally mounted with a permanent iron truss to the lower mast cap (the iron or wood fitting at the head of the lower mast). The former topsail yard became the upper topsail yard; it slid up and down the topmast on a parral, as in earlier generations. And by this time the lower yard also was fixed vertically by an iron truss.
The lower yard continued to be used as a derrick for handling cargo. In the case of the Cutty Sark (according to you-know-who's drawings), the lower yard lifts were heavy pieces of wire, leading from the lower yardarms through blocks at the lower cap. The end of each lift was hooked to a heavy rope tackle that, in turn, was hooked into an eyebolt in the deck at the foot of the mast. (My guess is that when the "yard and stay" operation was in progress for handling cargo, the sailors would take the hauling end of that tackle through the sheave at the base of one of the fiferail stanchions, in order to get more purchase on the line.)
The lower topsail yards don't have lifts. There is, however, a tackle called a downhaul on each side leading from the upper topsail yardarm to the lower topsail yardarm; that, in combination with the sail itself, would have kept the lower topsail yard horizontal. The upper topsail yards have "fixed lifts" - simple, one-piece wires running from the yardarms to the topmast caps. They go slack when the yard is raised (by means of the halyard) and the sail is set.
Lowering a yard to the deck, in a ship like the Cutty Sark, was a major operation that wouldn't be attempted at sea except in dire circumstances. The lower yards, in particular, were extremely heavy. (They were made of steel; the other yards were wood.) I doubt that the lower yards were sent down more than a handful of times in her entire career - and then only when she was tied up to a pier.
By the end of the nineteenth century the topgallants had been "doubled" as well. So in the last generation of square-rigged sail a large ship would have six sails on each mast: course, lower topsail, upper topsail, lower topgallant, upper topgallant, royal. (The Cutty Sark was built with double topsails; so far as I know she never "divided her topgallants." She did, however, initially have a main skysail above the main royal. The skysail was removed midway through her career - by the time the first surviving photos of her were taken.)
Again, the Campbell drawings will make most of this clear. (Rigging is far easier to explain through pictures than through words.) But I hope the above helps a little.