Without getting into the science of optics, there are a couple of pretty straightforward differences between zooms and primes.
The most basic difference, of course, is that a zoom lens can be adjusted to a range of focal lengths – you can zoom in and out – while a prime lens has a fixed focal length.
So, you may well ask, why would you ever want to use a prime lens?
Well, the next difference between zooms and primes helps us answer that question. All else being equal, a prime lens has fewer optical elements – fewer actual pieces of glass – inside of it than a prime lens.
Aside from making the lens smaller and lighter, which can be an important consideration under some circumstances, fewer elements means two things:
#1, the image is sharper. After all, looking through one window is clearer than looking through a stack of windows. Of course, a really good zoom lens may well be sharper than a really lousy prime lens, but generally speaking, the more pieces of glass you have, the more your image is going to get diffused.
#2, the lens if faster. Even clear glass absorbs some light. That’s why prime lenses almost always open up farther than zoom lenses. This is a fast zoom lens. It opens up to 2.8. But this prime lens opens up to 1.4. A full two stops more.
So. The advantages of zoom lenses are versatility and convenience. And the advantages of prime lenses are clarity and speed. Which you “should” use depends on what you’re doing.