Refractors have many wonderful qualities. Top-quality refractors are affordable by amateur astronomers in only the smallest sizes, up to about 6 or 7 inches or so, and even those are very expensive. The foregoing comments about the high quality images from refractors do not apply to the most common refractors seen, which are the cheap 60mm department-store refractors sold by Tasco, Meade, and a few other makers. These are typically of poor optical quality, come with impressive-sounding but cheap accessories, and have mounts of such poor quality that every year many budding astronomers are frustrated right out of astronomy by trying to use them. Do not buy such a telescope, no matter how tempting it may seem.
Reflectors gather light with a focusing mirror instead of a lens.
Because a mirror requires less exotic materials and is much easier to make than a lens, reflectors can be much less expensive than equal-sized refractors. This is a fairly straightforward procedure anyone can learn, but it's a nuisance which refractors almost never need. Catadioptrics are a combination of the reflector and refractor, sort of. They have a main mirror like a Newtonian reflector, but also have a thin lens in front. A similar type is known as the Maksutov-Cassegrain, a variant that is often a bit more expensive but often delivers sharper images.