Reflector: the reflector telescope consists of two mirrors named primary and secondary ones. The purpose of the primary mirror is to reflect the input light beam towards a smaller secondary mirror located at the input of teh telescope tube. There are three main diferent types of design depending on the curvature of the mirrors today available on the market: Newtonian, Cassegrain, Ritchey-Chretien.
The first design was invented by Newton and it is today the most popular telescope around amateur. The newtonian has a parabolic promary mirror reflecting the light towards a flat secondary mirror. This second mirror is positioned at the input of the tube in diagonal with respect to the tube itself. This allows the secondary mirror to reflect the light from the primary towards an eyepiece located at the input of the tube on its side.
The second design was invented by Cassegrain and consists of one parabolic mirror as primary one and an hyperbolic mirror as second. The primary mirror opreates like in a newtonian telescope reflecting the light towards the secondary one. This last one reflects and focus the light back towards a central hole located in the primary.
The Ritchey-Chretien is very similar to the Cassegrain but with a pair of hyperbolic mirrors instead of the parabolic/hyperbolic pair of the Cassegrain. The main principle is the same. This design is the most used today in professional observatories including the Hubble telescope.
Catadioptric: these are all those telescopes which operates by means of lenses and mirrors. The Schmidt Cassegrain, the Maksutov-Cassegrain and the Newtonian-Maksutov are part of the family.
The first design is definitely the most popular design in the last 10 years. launched by Celestron in the end of the 70's the Schmidt Cassegrain consists of an input glass corrector plate which corrects the spherical aberation introduced by a spherical primary mirror. Since the manufacturing of the corrector plate is not as difficult as theone of a large lens and a spherical mirror is much easier to produce than a parabolic one, this design allows mass production of low cost telescope with good performance and excellent portability.
The main difference between the Schmidt version and the Maksutov version resides int eh design of the corrector plate. Without entering too much in details this second desing is usually a little better from contrast point of view but slower because of longer focal length.
The Maksutov-Newtonian is becoming popular in these days; a correctorplate at the input of the tube helps the telescope to correct both for spherical aberration and coma. However with respect to the Maksutov Cassegrain, this telescope design features a smaller secondary mirror with consequent less aperture obstruction and then better contrast. I heard saying that this design performs as well as an apochromatic refractor.
4. What to buy now?
With all this offer it is not easy to go and purchase the telescope that is best for you. A very first suggestion is to absolutely avoid to buy telescopes from stores which sell you those 450X toys with plastic lenses and terrible eyepieces. Those telescopes are the worst enemy for any potential amateur for they are disappointing in terms of performanceand they are not even a good bargain.
We will analyze the kind of telescope by design including the type of mounting the telscope is going to use. This is a very important feature of the telescope and the mounting really affects a good 50% of the overall performance of the system
Achromatic Refractor: a good achromatic refractor must always be a long focal ratio telescope due to the better reduction in chromatic aberration obtained by slower lenses. Therefore we are talking here of telescope with a minimum f/8 focal ratio to guarantee a good color correction. This is definitely not a photographic tool if you are interested in imaging but it may be an excellent deep sky telescope when used with low power eyepiece. For planetary and moon observation, when magnification becorems important, the achromatic refractor performs pretty well but not so good as a simple reflector.
If you want to buy this kind of telescope my suggestion is to go for a minimum of 80mm aperture and f/8. The mounting may be either equatorial or altazimut but the second one is cheaper. Remember though that you need an equatorial mount to track the star with a motor. The price range for these telescopes is between $250 and $900 for the best ones.
A particular class of achromatic are the fast focal ratio wide field telescope. The main use of these optical tubes is often for bird sighting or other day activities. However these telescopes may be a good solution for wide field photograph of the sky. I have one of these and even the chromatic aberration is pretty strong due to the fast objective, the photoes you get are nice. These are usually in the 80-90mm aperture range and f/4 focal ratio. Their price is in the range $200-400.