Like most everything else in photography, lens come in various types, and can be categorized a few different ways. A simple lens, consisting of a single element, is rarely used for cameras, as they are subject to aberrations. most modern lenses are of the compound type, combing several different simple lenses. For example, the Nikon 20mm f/1.8 lens has six elements, while the AF-S DX NIKKOR 18-140mm f/3.5-5.6G ED VR has 17. Combining multiple elements reduces or eliminates aberration.
Lenses are described by two qualities: their focal length, usually in millimeters, and their maximum aperture, as an f/ number. The maximum aperture may be used to qualify a lens as fast or slow. So, a 50mm f/1.2 is fast (or faster) than a 50mm f/2. The focal length represents the distance from the lens to the focal plane (film plane) of the camera when focused on infinity. More importantly, it represents the angle of view of the lens. This is the first major division of lenses, by their angle of view.
A normal lens will project an image the size of the diagonal of the film (or sensor). This will also approximate our human angle of view, somewhere around 45º. So the lens sees pretty close to the way we see. As formats change, the length of the normal lens changes. For 35mm format, normal is about 50mm. For medium format, around 80mm, and for 4 x 5 format, 150mm. The normal lens will produce some distortion of subjects that are close.
Wide-Angle, or Short Lenses
Short lenses are those with focal lengths less than normal for a given format. They are also called wide-angle, as their angle of view increase, as the focal length decreases. For the 35mm format, a 35mm lens has about a 60º angle of view, while something like a 16mm lens would have up to a 180º angle of view. The medium-short lenses are fairly popular, with their wide view and relatively low distortion, while the very short lenses produce great distortion, fish-eye type. Wide-angle lenses also have shorter minimum focusing distances than normal, as well as a longer depth of field at the same aperture as a normal lens.
Telephoto or Long Lenses
These lens have focal lengths longer than normal, again for a given format. Telephoto lenses have less angle of view than a normal lens, with less depth of field at a given aperture. They also compress relative distance, making objects in a scene look closer together, near to far, than they actually are. These lenses, for 35mm format, range from 60mm to a somewhat unique 1200mm made by Canon. Of particular interest in this category is the "portrait" lens, which is a shorter telephoto that can photograph a person's head without the distortions of a normal lens. For the 35mm format, this is around 90mm, and around 120mm for medium format.
The term prime simply means a lens of a fixed focal length.
Macro (or Micro)
These lenses are designed for close-up images, with a higher magnification ration than a standard lens, typically 1:1. They have longer barrels (usually) which allows a closer focusing distance, as close as 6 inches. They will usually have more elements than their standard counterparts.
Zoom lenses are multiple focal length lenses, able to be used over a range of focal lengths, as opposed to primes. They can be short, long or both. They may cover a smaller range, like a 24-70mm, or a much greater one, as a 28-300mm. The quality of the image may be debatable over a prime of a given focal length, but zooms can be very good, and their convenience should be self-evident. A medium-range zoom, in the 50-120mm range, is excellent for photographing artwork, for instance, as it means moving the tripod less for differently-sized images.
Tilt/Shift, Perspective Control Lenses
These specialty lenses have the ability to tilt and shift the lens, breaking the parallel relationship between the lens and film plane. Their main use is to correct parallax error, or converging parallel lines, particularly in architectural photography. They allow view camera-like movements with an SLR.