I'd say you might want to read
this lens primer, which will give you some basics on what lens descriptions mean in practical working terms. There is no set number of "types" of lenses, as a lot of the different lens characteristics can be combined in different combinations.
In terms of some basic types of lenses, though:
walkaround zoom
Basically means a lens with a variable focal length that serves most general purpose shooting, like portraits/landscapes, and general snapshots.
portrait lens or
fast prime
Means a non-zooming lens with a very large maximum aperture. This has the effect of blurring the background, and to shoot in lower light capabilities than lenses with smaller max. apertures.
telephoto zoom
Means a lens with a variable focal length that has a lot of "magnification". It's typically used for things like sports or wildlife shooting--anything where your working distance from the subject is likely to be large.
normal or
standard
is a lens that is neither wide angle nor telephoto. Traditionally with film photography (and full-frame digital cameras) this is a focal length ~50mm. What you see through the viewfinder matches the same magnification as your naked eye.
wide angle or
ultrawide
Are lenses that do the opposite of what telephoto lenses do. They make you feel farther away from the subject, and give an expensive field of view. Wide angles and ultrawides are typically used for landscape photography, or for working in close or small spaces.
macro
All lenses have a minimum focus distance--that is, the nearest to the subject the lens can get and still achieve focus. Macro lenses are specially designed to have the ability to focus on very close objects. Macro typically means that the lens can achieve 1:1 magnification--that is, the size of the image on the sensor is the same as the actual object size.
In terms of what to look for when buying one, you need to know three major things, first:
- Do you really need a lens or is it your technique?
- How much can you afford?
- What do you want to use the lens for?
Generally, if you're just starting out, the advice I tend to give is to just stick with the kit lens until it frustrates you in some way. The way it frustrates you will probably guide you to what you want out of your next lens, or what technique you may need to work on.
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Addendum, re: costs. If you're looking at buying brand new lenses, "cheap" is essentially $300 and under. "moderately expensive" is from $300-$1000, and "expensive" starts at $1000 and the sky's the limit. At least that's the way I personally categorize how the camera companies think. You can look up prices on Amazon, B&H, or Adorama to get an idea of how much lenses cost.