What is so great about a prime lens, vs a standard 18-55mm lens?

Generally speaking, a prime lens is going to have the best optics. The elements are optimized for that focal length, while a zoom has to be a compromise over a range of focal lengths. Also, if you are talking about a lens that has a f/stop of 1.8 or 1.2 vs a zoom with a range of 3.5-5.6, the prime lens lets in more light, so you can shoot at faster shutter speeds or really make the background blurry, giving you creative control.

There are excellent zoom lenses out there, the Canon L series of lenses are great examples, but the kit lenses that come standard with most of the entry level dSLRs are just decent at best. Good to get you going, but if you want to really craft excellent photos, you need to upgrade at some point.

The fastest prime I have is a 50mm f1.4, and the fastest 50mm lens for an SLR camera is f/1 (Canon 50mm f1.0 L USM). Your 18-55mm lens is f/3.5-5.6, which at 50mm is near as makes no difference f/5.6, that makes it 5 stops slower than the fastest 50mm.

Additionally to make a zoom lens the engineers need to make the lens at a continuous range of focal lengths from 18mm all the way to 55mm, this does often affect the maximum sharpness achievable, as well as causing vignetting, barrel distortion and CA problems. A prime (especially a 35mm, 50mm or 85mm) is very easy to make, and require relatively less glass. As such a prime will almost always be sharper, and cleaner at all apertures that the zoom can handle.

The drawback of the prime is obvious, it is very rigid, and if you can't fit/fill a frame you've got to walk back/nearer to achieve what you want, however sometimes this is physically impossible (e.g. a barrier, or a wall in your way). In certain situations a zoom can give you flexibility, for example at an event, you may be shooting a mix of groups and candid portraits, this means you need a wide angle lens (28mm on full frame) and a moderate telephoto lens (85mm) if you were shooting primes, this means you're more likely to miss certain shots. This situation having a 24-70mm f2.8 zoom is will give you a much higher hit rate (of course at great cost, and some sacrifice to DoF control and IQ).

If you know what you will be shooting at a given job, it's best to use primes only. If you haven't a clue what may happen (which is quite regular for a journo), then pack the zooms. And in most cases you have some idea of what's going to go down, pack your primes, and a zoom just incase something interesting happens.

There are good prime lenses and bad prime lenses, just like with zooms.

The 18-55 kit lens, is quite literally the worst quality lens you can have. Just about any lens is going to be sharper.

Prime lenses are popular because you get a lot of bang for your buck. The 50mm 1.4 is incredibly sharp for a $350 lens. To get the same optical quality in a zoom, you would be looking at something like the 24-70 f/2.8 which costs closer to $1,500.

However, I think anyone who thinks that a lens is better just because it is a prime doesn't know what they are talking about. There are plenty of excellent zooms and shitty primes out there. But dollar for dollar, a prime will tend to be better.

I wouldn't say they are "the best", but image quality tends to be better with a prime lens (though nowadays zoom lenses have narrowed the gap in quality). Additionally, prime lenses tend to have a wider maximum apertures (giving you more narrow depth of field) than their zoom counterparts.