A 'complication' is watch-speak for any function a watch does beyond showing the hours and minutes. Some are everyday; some are feats of engineering that take a master watchmaker months to build.
A tourbillon puts the escapement and balance wheel inside a tiny rotating cage. Invented around 1801, its original purpose was to average out the effect of gravity on a pocket watch that sat in one position all day. On a wrist that moves constantly it is less necessary, so today it survives mainly as a mesmerising showcase of skill — which is why tourbillon watches are among the most expensive.
Date and day windows; a chronograph (a built-in stopwatch with pushers); a GMT or dual-time hand for travellers; and a power-reserve indicator showing how much wind is left.
The summit of the craft: the perpetual calendar (which knows month lengths and leap years for decades), the moonphase, and the minute repeater, which chimes the time on tiny gongs. A watch combining several of these is called a 'grande complication'.