They are broken into major classifications with many variations due to professional, amateur, and commercial sub-types.
Telescopes can be classified by optical design or mechanical design/construction.
One major determining factor is type of light, or particle being observed including devices referred to as "telescopes" that do not form an image or use optics.
Optical telescopes can be classified by three primary optical designs (refractor, reflector, or catadioptric), by sub-designs of these types, by how they are constructed, or by the task they perform.
They all have their different advantages and disadvantages and they are used in different areas of professional and amateur astronomy.