A serpentine belt is an all-in-one device that drives a number of accessories and critical parts on a modern automobile. It takes the place of a number of separate belts that were used in earlier car models, and are still used today in some cars: fan belt, alternator belt, and 'accessory belts' used to power in-car accessories, like your radio and other components. All of these belts work in basically the same way; they take power from the engine when it is running to power other devices in your car, or synchronize timing through a system of toothed belts and gears. Since all of these belts do basically the same thing (pull RPM's off the engine and use them to generate motion, either of an actual rotating part (fan) or of a generator (in the alternator or power steering pump)), someone got the bright idea of taking one big, long belt and looping it through every single device in the car that runs off a belt.
Since it winds around all over the place, it is called a serpentine belt. The advantage to the serpentine belt is that it reduces the number of points of failure in the car; no longer can any of 15 belts break or come loose. The problem is that the serpentine belt then becomes one massive source of failure; while you may be able to limp along in a multi-belt car that is missing or slipping a belt or two, a problem with the serpentine belt generally means that the car does not go, or at least does not go in a manner that makes you want to keep driving it. Common problems with a serpentine belt are a loose belt, which causes a terrible screeching noise, or a broken or slipped belt, which can cause your car to not go, or can cause you to lose control of the car if your power steering pump looses power while you are attempting to drive.
Heat and cold take their toll on the serpentine belt, like any belt, as well as regular wear and tear. This is the cause of most belt slips or breaks, and thus most problems stemming from the serpentine belt. Most mechanics recommend that the serpentine belt be replaced as part of regular maintenance, at least by around 90,000 miles (that's an 'it should be OK' value, rather than the maintenance your manual suggests. YMMV). In addition to normal wear and tear, problems with a serpentine belt can be caused by a defective belt tensioner, which is responsible for- you guessed it- keeping the belt at the right tension, so as to prevent slipping or sqealing. For further information, consult Click and Clack.