As Laxx said, aperture is the main advantage prime lenses have over zooms. Discounting the two nutball F/2 zooms for the 4/3rds system, the fastest zooms are F/2.8. Most zooms start at F/3.5 or F/4 and shrink down F/5.6 at the long end. Constant aperture zooms, especially F/2.8 ones, are quite expensive. Wider apertures are possible with prime leneses, but they can be pricey too. Canon's 50mm F/1.8 is under $80, but it's built like a toy. The F/1.4 version is more than four times that. The 50mm F/1.2L is $1200, and IMJO, complety not worth it. The long-discontinued EF 50mm F/1.0L sells for more than an HE90 on the used market.
There are some very good primes that aren't bank-breakers. The EF 35mm F/2 ($250) is the best deal in the entire Canon lineup. The EF 85mm F/1.8 is also a steal, especially for photographers overly concerned with bokeh. If you don't know what that is, consider yourself among the sane. Primes have several other optical advantages, they are often sharper than zooms, and usually have less distortion. Usually they are smaller and lighter as well.
Prime lenses are popular with skinflints, available light shooters, and pretentious photography snobs. I happen to be all three. There is something to be said for the fixed field of view a prime offers making one a better photographer -- or at least offering the potential for such growth.
Most very wide and and very long lenses are primes, though recently that has changed on the wide end (Nikon's 14-25mm F/2.8 is a really groundbreaking lens.) Lastly, there are all sorts of special purpose prime lenses: macros, fisheyes, tilt/shift, soft focus, etc etc.