I've been reading lots of camera reviews lately and have ran across several references to "fast" lenses.
What exactly is a fast lens and what are its advantages compared to other lenses?
Ysap and Matt gave great answers. I understand things best with metaphors, so here goes.
Consider a container that you wish to fill with water. In this case, the water will represent light, and the container will represent your lens. The wider the opening in the container (the aperture), then the faster you'll be able to fill the container with water (light).
If you were to take two containers, one with a very large opening and another with a very small opening, and dump the same amount of water onto each of them (for the same amount of time), which container would be filled with more water? Clearly the one with the bigger opening. If you wanted both containers to end up with the same amount of water, then the one with the smaller opening would have to have water dumped on it for a longer period of time than the one with the larger hole.
It works the same way with lenses. A camera with a small F-stop is a lens that can open up very wide, thus having a bigger opening to swallow up light. What this means to a photographer is that he/she can use much faster shutter speeds and still get adequate light in his or her photo. In this case, consider a swimmer coming up very briefly for a breath of air. The swimmer would much rather open his mouth up wide to take in air than purse his lips and try to breath in. A fast shutter speed with a wide aperture means more light can get in during the short period of time that the shutter is open.
Consider shooting a ballet in a dark theater. It would be best to use a lens with a wide aperture, because of the dark environment. Additionally, you would be able to use a faster shutter speed to capture the fast-moving dancers.
I hope this makes sense!
What matters for distinction between fast and regular lenses is relative position of maximum aperture in regard to general availability of lenses for same format and focal length. For any given lens, if one can't easily (ignoring price) buy a lens of similar focal length for same image circle size with maximum aperture more than a stop wider, it's a fast lens. I'm not aware of this being written down as a rule anywhere, but that's how the distinction seems to be made in practice.
So in 35mm/APS-C world, 50mm f/2 is considered not very fast (with the f/1.8, f1.4 and f/1.2 around), but a 600mm f/4 is fast. On medium format, 50mm f/2.8 would be a fast wide-angle lens.
Zooms are more complex and cannot usually reach similar apertures as primes, so they have to compared to other zooms in similar focal length range for same format. For zooms with variable aperture, usually the aperture of longer end is used in comparison. So although a kit zoom may be f/3.5 in its wide end, not far from professional f/2.8 zooms, the f/5.6 in the other end will keep people from calling it a fast lens.
A fast lens is a lens with a wide maximum aperture (that is, a small F number). This lens lets you use faster shutter speeds for a given amount of light. For example, EF-50mm/F1.4, EF-70-200mm/F2.8.
I have seen once or twice (don't have an available quote) people use the term "fast" in conjunction with fast focusing mechanism (which, many times, is constructed in a fast lens). These focusing mechanisms usually consist of an Ultrasonic Motor (USM) of some sort and tend to focus faster than those based on DC motors. Note that this is NOT the regular meaning of the term "fast lens" and should be avoided, as it may confuse your audience.
A fast lens is one which has a wide maximum aperture.
The aperture is the opening which controls how much light reaches the film or sensor, and lenses have diaphragms (composed of "aperture blades") which can open up or "stop down" to control the exposure. This is usually done in pre-defined steps, called f-stops. The standard terminology uses smaller numbers (like f/1.4 or f/2) to indicate a wider opening, and higher numbers (like f/22 or f/320 to indicate a smaller one. A wide aperture (small numbers) lets the same total amount of light in more quickly than a narrow aperture (big numbers), so it's faster.
The widest possible aperture on a given lens tells you how fast it is. There's no hard rule, but generally f/2.8 is considered fast for a zoom or extreme telephoto prime lens, and wider than that (f/1.8, f/1.4) for a normal prime. Some people might consider f/2 borderline for a 35mm or 50mm prime lens.
Fast lenses have several advantages:
The disadvantages are the normal compromises for lens design, mostly centered around the fact that a wider aperture requires larger glass. That means larger overall size, more weight, and greater cost. Additionally, when a lens is optimized for shooting at fast apertures, the design may trade off general performance. Or, as is often the case with older lenses, faster apertures may be available but several image quality factors (sharpness, contrast, light falloff) not at their best levels until you've stopped down quite a bit.
Also note that many cheaper zoom lenses (including, as far as I know, every built-in zoom on compact cameras) have a variable maximum aperture over the zoom range. That means the lens might be relatively fast when zoomed out but not fast at all zoomed in.