These lenses use a mirror in their rear surface to reflect the image to another mirror at the front surface, which then finally reflects the image to the camera sensor. This effectively folds the focal length in half, enabling mirror lenses to be much smaller, lighter, and cheaper. Mirror lenses, because of the central obstruction in the front, always have less contrast than a good refractive lens. They also generally have less sharpness than traditional refractive lenses, in addition to a fixed aperture (usually F-8 to F-11). Mirror lenses tend to provide a more confused and distracting background (bokkeh as it's called) than refractive lenses, with lots of false detail. Refractive lenses provide a much smoother out of focus region.
Interestingly, mirror lenses at an equivalent focal length are less prone to camera shake because they are MUCH smaller and have MUCH lower mass. You also won't need hired help to cart them around.
But most importantly to me, mirror lenses are MUCH!!! cheaper then equivalent refractive lenses. Nikon's latest 600 mm AF-S lens will set you back $12,000. I bought a 1000 mm F-11 Celestron lens on ebay for $150 bucks, and it takes GREAT astrophotography shots, even with a 2X teleconverter attached. Click on the first picture of the moon to see the full resolution. Yes, it’s manual focus, but so what? With the 2X teleconverter and the Nikon sensor size (which magnifies everything by 1.5X), I have an effective focal length of 3000 mm for cheap!!! This fills up the entire frame with moon! Also, such lenses are very useful for photographing birds as well as other small and distant objects. It's just a ton of fun to play around with, even if it doesn't give you the photo quality of a $12,000 lens.
