Typically very inexpensive and not very fast or especially sharp, kit lenses are sold in camera kits to sweeten the deal of a camera purchase. A great place to start for general-purpose photography, you may eventually find that you want a broader focal range, a faster maximum aperture or higher-quality optics. But, for starters, a kit lens is fine for most photographers.
A prime is a fixed-focal-length lens, such as 35mm, 50mm, 85mm and so on. Typically faster (i.e., possessing a wider maximum aperture, smaller ƒ-number) due to the simplified optical construction, primes are frequently smaller and lighter than zoom lenses, as well, and traditionally are high quality, by default. Professional models are larger, but also exceptionally sharp.
A zoom lens can be adjusted across a range of focal lengths, for instance, a 24-70mm zoom lens. Some zooms are wide-angle, some are telephoto, and some “extreme zooms” cover a range from wide to telephoto. As for benefits, a single lens can cover a wide range, so a single zoom can replace two or three prime lenses. This makes it ideal for travel or street photography, where the ability to change focal lengths quickly and to travel light are distinct benefits. One downside is that fast zooms (with wider maximum apertures) are very expensive.
A normal lens, in 35mm film terms, is a 50mm lens. This lens approximates the angle of view of the human eye (or so it was long agreed, even though the human eye sees things with a slightly narrower angle of view), so it’s considered to produce a normal view. Normal lenses range, generally, from 40mm to 60mm in 35mm equivalent terms.
A wide-angle lens is generally any lens shorter than 40mm, which provides an angle of view “wider” than a normal lens. Typical wide-angle focal lengths are 17mm, 20mm, 24mm, 28mm and 35mm. Lenses shorter than 17mm are considered “superwide” and provide a huge angle of view up to and beyond 180 degrees. The widest of these lenses are known as fisheyes.
A telephoto lens has a focal length greater than 70mm in 35mm equivalent terms. Popular telephoto focal lengths are 70-200mm zooms, and 85mm, 100mm, 135mm and 150mm primes. Especially long lenses, above 300mm, are called supertelephotos.
A lens ideal for portraits is a short to medium telephoto with a wide maximum aperture. So, a 100mm ƒ/2 lens, for instance, or an 85mm ƒ/1.4, are ideal portrait lenses. They allow for compression to simplify backgrounds and minimize facial features, while not requiring you to stand so far from your subject that you can’t easily interact with them.
A sports lens, also good for wildlife, is a supertelephoto such as a 400mm prime or a 100-400mm zoom. High-end lenses such as 600mm and 800mm primes were all but mandatory for professional sports and wildlife photographers until resolution advances in the digital era, combined with crop sensors that make focal lengths effectively longer, made extreme telephotos less necessary. Today, a 400mm lens is plenty for most sports and wildlife needs.
If you hear someone refer to a landscape lens, they’re typically referring to a wide-angle lens, maybe even especially a prime lens, that allows for foreground/background compositions popular in landscape photography. A 35mm lens is a great landscape lens.
An architectural lens incorporates movements made popular by large-format view cameras generations ago. These movements—tilt, shift, rise, fall—allow the focal plane to be manipulated and for the correction of distortion that occurs when a wide-angle lens is aimed up at a building. These lenses are also called tilt-shift lenses and perspective-control lenses by different manufacturers. Typically, these lenses are wide-angles (17mm, 24mm), although some brands offer normal and even telephoto lenses with tilt-shift controls.