Refractors use a lens system to produce images.
Refractors require the least maintenance of all telescopes.
Although all telescopes produce an inverted image, refractors use a star diagonal to turn the image up the right way, although it will be flipped left to right.
Refractors are either Achromat or Apochromat. Apochromats, which are generally more expensive, do not suffer from false colour.
Reflectors use a system of mirrors to produce an image.
A reflecting telescope offers the best £ per inch of aperture ratio - a large mirror is cheaper to make than the equivalent sized lens.
The largest amateur telescopes are reflectors.
Reflectors are not suitable for terrestrial viewing as the image will always be inverted.
Catadioptric or compound telescopes, use a combination of lenses and mirrors to produce the image.
They are the most compact design, giving shorter tubes that are typically 1/4 the length of reflectors and refractors.
Catadioptrics come in two designs - Maksutov-Cassegrain (Mak's) and Schmidt-Cassegrain (SCT's). SCT's tend to be more expensive.
Catadioptrics use a star diagonal, so are therefore suitable for terrestrial viewing.