Types of Telescopes
There are two basic types of telescopes: refractors and reflectors. Both have their advantages. Properly outfitted, either will show you distant galaxies, the rings of Saturn, or the craters of the Moon.
Refracting telescopes gather light with a lens, directing it to the eyepiece. Small starter scopes are often of this type, as they are simple to operate and maintain. Larger refractors, however, become very expensive and are typically bought by avid enthusiasts.
Reflecting telescopes gather light with a mirror, reflecting it before directing it to the eyepiece. Reflectors typically give you a wider aperture for the dollar. They require more care and maintenance, however.
The simplest type of reflecting telescope is called a "Newtonian," after Sir Isaac Newton who invented them. When a Newtonian telescope in it's simplest and most user-friendly form, it's called a Dobsonian telescope after John Dobson who showed the world that you don't have to be rich to make your own telescope.
Probably the most common type of serious amateur astronomical telescope is the Schmidt-Cassegrain telescope. It is a reflector telescope that also uses a corrector plate to eliminate chromatic aberation.
A second type of hybrid, called a Maksutov-Cassegrain, is renowned for the views it offers. They are difficult to make and are not as common as Schmidt-Cassegrains. They are popular for the new Meade Autostar telescopes.
For more information about how telescopes work, here's a link to Meade's excellent resource!