Prime lenses are often smaller and more compact than zooms, and tend to be faster, offering wide apertures of f/2.8, f/1.8 or even f/1.4. These large apertures let in more light, which benefits the photographer in a number of ways. First, you can use fast shutter speeds with large apertures to freeze motion. This becomes important when you're shooting in low light. By using a high ISO and a wide aperture, you'll be able to use high enough shutter speeds so you can hand-hold the camera. Secondly, because fast lenses let in lots of light, the viewfinder will be brighter, and so it will be easier to confirm focus, ensuring your images will be sharp.
Also, as primes are often simpler in construction than zooms, this means they are not only often slightly less expensive to make, but they also tend to have fewer visual aberrations such a colour fringing and image distortion, giving you a more technically correct image than a zoom can at the extremes of its focal range
Another popular benefit of a prime lens is the bokeh it will achieve, especially for portraits. Bokeh is the pleasing, diffused softness of the blurred background in an image with shallow depth of field. And then there's the fact that with a prime, to make your subject larger or smaller in frame, you actually have to move towards or step back from it when shooting, as you can't zoom in and out from a static position as you can with a zoom lens. Many photographers appreciate primes for this exact reason, as they say it encourages them to be more creative in their picture making – it certainly makes you think more about what you're shooting.
Of course, with any piece of kit there are always compromises, and with primes they tend to be twofold: first, whereas you can cover off a range of focal lengths in a single zoom, you might need several primes to do the same, and secondly, they can mean missing out on fast-happening photo opportunities, whereas with a zoom you do have more flexibility. Ultimately, it depends on what you want to shoot and how you like to work. But for many photographers, prime lenses' stunning image quality and speed, especially in low light, usually make them an indispensible part of their kit.