Owners of DSLR cameras have a dizzying array of lenses available for their cameras (that's sort of the point, after all), but choosing the right lenses for your bag can be a challenge -- especially when you're on a tight budget. Among the biggest choices is whether you want a zoom or a prime.
Prime lenses are the simplest lenses you can buy. They have only a single focal length - 50mm, for example. This means that if you want multiple focal lengths, you must buy a lens for each focal length. The diagram here (from http://www.canon.com/camera-museum/) shows the optical elements in a Canon 50mm f/1.8 prime lens, a very popular prime for beginners. The mechanism here may look complicated, but it's far simpler than a common zoom lens.
Zoom lenses let you use one lens to cover multiple focal lengths, but in order to do that, they're more complicated optically and mechanically. The next diagram shows the inner workings of a Canon 17-85mm f/4-5.6 IS USM lens, which is a fairly common mid-range Canon zoom. These comparisons are going to vary from lens to lens, obviously, but you'll find as a general rule, zooms are just a little more complicated.
As a result, primes are sharper than zooms, all things being equal. If you take a really high-end zoom, for instance, it might be able to give a cheap prime lens a run for its money, but if you compare two lenses with similar-quality build and optics, the prime will be sharper than the zoom, and probably more optically true as well (less distortion).
The "compromises" in a zoom lens also tend to be more accentuated with greater zoom range (ie, it's easier to make a high-quality 18-55 lens than an 18-200 lens), so you don't find lenses that have huge zoom ranges *and* good quality *and* a low price.
One last factor where primes tend to have a huge advantage in in maximum aperture. The aforementioned 50mm f/1.8 lens is commonly available in the neighborhood of $100, but the fastest zooms you'll find have a maximum aperture of f/2.8 (and you won't find any of those for $100). Since it's so much more difficult to build zooms with large maximum apertures, you tend to see stabilization technologies in zoom lenses, but not in primes.
With all these factors seemingly in favor of primes, why would anyone ever buy a zoom? Simply put, you just can't cover the range of a couple of zooms without buying a ridiculous number of prime lenses, so in almost all cases, the ultimate solution is a combination of zooms with a prime or two in focal lengths you use most, or in portrait lengths where you want the best possible quality.
Of course, this is a *very* general discussion, and you should definitely research any specific lenses you're interested in before you buy them, but hopefully this will get you started in the right direction.