A lot goes into the average engine - most cars have at least 10,000 parts, and engines account for quite a few of those parts. Some parts, like the onboard computer, are very complex and require specialized equipment just to interact with them. But other parts are much more simple, and yet are equally critical to the operation of the vehicle. The belts that support various aspects of the car fit that description - in concept, automotive belts are very simple. But if not maintained, failing belts will bring any vehicle to a halt.
Different Belts for Different Needs
Different types of automotive belts operate in different parts of the car, and serve different functions. However, even with all of the terms out there, there are really two primary kinds of belts used in the average car. In older or classic cars, you may encounter more belts, as several additional belts used to power various component. In the modern car, those have largely been replaced by the Accessory (or Serpentine) belt.
Serpentine belt/Accessory Belt/Drive Belt
Typically, this belt is referred to as either a drive belt, accessory belt or serpentine belt. Basically, they're all referring to the same belt. The 'serpentine' actually refers to the configuration of the accessory belt in many cars, and 'V-belt' or 'Poly-V' belt refer to how the belt itself is constructed. The correct term for the belt is an accessory belt, because its primary function is to drive those items in the engine that are considered accessories to the primary engine - namely, the power-steering pump, the compressor for the air conditioning, the water pump (usually), and the power-steering pump. The accessory belt may also power other parts as well.
The accessory/serpentine belt looks like it sounds - it's typically visible on the exterior front of the engine, and it twists and turns like a serpent (hence the name) around several pulleys located around the engine. The accessory belt is a very important component to any engine, and they're built to last - these days, the average accessory belt with last anywhere from 60,000 to 100,000 miles before they need to be replaced.
The other critical belt in the modern engine is the timing belt. Without the timing belt, virtually every engine will fail almost immediately, and won't start again until the belt is replaced (assuming no other damage has occurred).
Basically, the timing belt provides an interface and coordination between the crankshaft, which takes the explosive energy produced by the pistons and converts it into rotational energy that can be used by the rest of the car - and the camshaft, which is the component that regulates the opening and closing of the engine's valves to move air and gas in and out of the engine.
The timing belt, then, coordinates the timing between those components, to ensure the smooth flow of energy throughout the engine. Should the timing belt fail, that coordination fails, followed shortly by the engine shutting down.
The timing belt is often not as easily accessible as the accessory belt, and in some cars can be difficult to reach. But it's usually at least somewhat accessible, for inspection if nothing else.
As mentioned above, in some cars there may be more than two belts. This is especially true of older classic cars, but additional belts may be found in some modern cars as well. Typically, these belts are other accessory belts that, due to the design of the engine, power components that couldn't be accessed by the primary accessory/serpentine belt. If such a belt is present, consult the car's owner's manual for more information.
Looking for Trouble
Because both belts are so critical for successful car operation, it's important to routinely inspect both belts for signs of wear. If possible, it's always a good idea to catch problems with either belt before they fail, because failure in either of the belts can produce larger problems, and damage, to other components in the car. This makes what would have been a relatively inexpensive belt replacement into what can be major engine problems.
Another good idea is to look for tension on the belts, especially the accessory/serpentine belt. Most serpentine belts, due to their long length and non-linear shape, require a component called a tensioner to be located somewhere along its run. The tensioner does just what it sounds like - it provides tension on the serpentine belt to keep it running tight and aligned. Tensioners are like any other part in a car, and they can run down over time and mileage. If the serpentine belt seems to be sagging or lacking tension, the tensioner should be inspected.
The accessory/serpentine belt can typically be replaced using hand tools, the right parts, and a little knowledge. The timing belt, however, should usually be replaced by a mechanic. However, by regularly checking both belts proactively and regularly, both belts can be replaced relatively inexpensively, avoiding much larger problems down the road.