What kinds of telescopes are there?
The traditional answer for optical telescopes is reflectors (use mirrors) and refractors (use curved lenses). There are a few various mounts and styles as well, that could be considered different “kinds.” There are also telescopes that measure parts of the EM spectrum that is not the equivalent of visible light: radio telescopes, infrared telescopes, etc. For personal “backyard” astronomy, the answer you are probably looking for is reflector and refractor.