Zoom lenses are almost ubiquitous now but fixed focal length, or ‘prime', lenses continue to be popular because they offer several advantages. Since they only have to convey a single field of view, as opposed to having to offer a variable range, the optical performance is generally superior. They tend to be smaller too, and as an extra bonus have wider maximum apertures. If the field of view they offer is the one that you want, and you can fine-tune your cropping simply by moving your position, then they have a lot to offer.
Primes cover the full spectrum of focal lengths from extreme wide to ultra telephoto, but the most useful are those in the 24mm to 85mm range. Years ago most 35mm SLRs came with a 50mm standard lens, as its field of view most closely resembled that of the human eye. Nowadays most DSLRs have smaller APS-C sensors so it's the 35mm focal length lens that now has that distinction. Nevertheless, 50mm lenses still offer the widest available maximum apertures (as low as f/1.2) making them ideal for low-light photography, especially when you have to shoot handheld, without flash, possibly with subject movement to contend with (such as at concerts). Longer focal lengths such as the 85mm, are perfect for portraits, offering a flattering perspective and the ability to throw the background nicely out of focus.
Prime lenses tend to have wider maximum apertures than zooms, making them ideal for low-light shots, especially of moving subjects such as this rock band (left). For portraits it's hard to beat a prime telephoto such as the 85mm f/1.8 used on a Nikon D700 for the shot above. The shallow depth of field they afford is perfect for throwing the background out of focus.
Lens Top Tip
The 50mm f/1.8 is the bargain of the lens world. Small in size, with a wider maximum aperture than any zoom, it's one of the cheapest lenses you can buy, and yet optically is among the best. Everyone should have one!