Unlike refracting telescopes, which use a lens, reflecting telescopes use a combination of curved mirrors to produce an image which can the be magnified using an eyepiece. The first reflector was designed by Sir Isaac Newton (of apple on the head and gravity fame) in 1668. This simple and elegant design is still in use today.
The biggest advantage reflecting telescopes have over their refractor cousins is size. The largest refractor in operation is a 40 inch professional scope at Yerkes Observatory in Wisconsin. However the largest single mirror reflector is a whopping 323 inches! Of course these are professional instruments.
For us amateurs the big advantage reflectors have is price per inch of aperture. You can find a nice 6 inch Newtonian reflector on a decent quality equatorial mount for around $500. A fair refractor on the same mount would cost at least $1000. You can buy some nice eyepieces with 500 bucks!
Another plus on the reflectors side is their lack of false color. Refractors are notorious for halos of color around bright objects. This is due to the physics of lenses, the different wavelengths of light focus at different points. This can be mostly eliminated in modern apochromatic refractors, but this can add greatly to their price as well. For example my 90mm APO cost almost $1000. Beautiful scope, and you couldn't pry it out of my fingers but still...
Of course the basic design of Sir Isaac has changed over the years and now there are many different types of reflecting telescopes available to both professional astronomers and the amateur community.
