Reflector lenses (also called mirror lenses, or catadioptric lenses) use one or more mirrors as optical elements, in addition to lenses. This allows the optical path (i.e., the path followed by light passing through the lens) to be folded. Typically, mirror lenses have two mirrors. Light enters through the front element, is reflected by the first mirror at the back of the lens barrel, travels to the front of the lens and converges onto a smaller mirror located on the front element (usually hidden from view behind a circular disk or baffle at the centre of the front lens element), and from there it is reflected back, passes through a hole at the centre of the first mirror and reaches the focal plane. Folding the optical path allows a mirror lens to be very short relative to its focal length. Typically, mirror lenses have a focal length of 500 mm or more in a lens barrel only about 150 mm long. If you look through the front of a mirror lens, you can see the surface of the first mirror. If you look through its back, you will see the second mirror.
Mirror lenses are cheaper to build than comparable lenses that use only refractor optics (i.e., transparent lenses). However, most mirror lenses also have a lower contrast and sharpness than more expensive refractor lenses (see my tests here). The Olympus OM-System Zuiko Reflex 500 mm f/8 is one of the exceptions. Mirror lenses also have several other undesirable characteristics. The principal ones are the lack of a diaphragm (so they have only a fixed aperture, typically f/8 or f/11) and the rendering of out-of-focus bright points as doughnuts.
A makeshift diaphragm can be made with a black cardboard disk cut to fit the filter mount in front of the lens. A smaller circle is cut out in the centre of the disk (obviously, it has to be wider that the central baffle covering the second mirror). This "diaphragm" does decrease luminosity and increase depth of field, but makes the image in the viewfinder even darker and more difficult to focus on manually. There is no way, on the other hand, to eliminate the doughnut-shaped blobs in out-of-focus areas.
Usually, mirror lenses also lack autofocus, and with some cameras they do not allow automatic exposure.
In spite of these characteristics, mirror lenses typically cost 1/5 to 1/10 the price of good refractor lenses of similar focal length, and may be the only long telephoto lenses that some non-professional photographers can afford to buy (or justify buying). If you can afford a refractor telephoto, by all means choose it instead of a mirror lens. However, if you cannot afford the latter, a mirror lens is a valid introduction to super telephotography. After all, you cannot shoot with a lens you don't have, so a mirror lens is better than no lens at all.
It is possible to reduce the aperture of a mirror lens by placing a black paper or cardboard sheet with a circular hole at its centre directly in front of the lens. This, however, further reduces the luminosity of the lens and makes manual focusing even more difficult. It may also cause vignetting or light fall-off at the edges of the picture.
Some enthusiasts use reflector telescopes instead of mirror lenses designed for cameras. These telescopes have the same basic constraints as mirror lenses. In addition, these telescopes are designed to be very sharp in the centre of the picture, but resolution may fall off rapidly toward the edges.
There are special designs of mirror lenses that eliminate one or more of their characteristic drawbacks. However, they are difficult and expensive to build, and have remained novelties, rather than practical designs. Mirror lenses are also used in specialty microscope objectives, for instance when an objective capable of transmitting from the infrared to the far ultraviolet is needed. Since these special lenses are built without refractive (i.e., lens) elements, light does not pass through glass, and infrared and ultraviolet light is not absorbed (as long as the mirror coating is effective at these wavelengths). Mirror optics are also used in special microscope condensers, and in all large astronomical telescopes.
For a few years in the past, Vivitar made "Solid Cat" lenses. They are reflector lenses in which light travels through a solid block of glass with front and back mirror surfaces, rather than through air between two mirrors. These specialty lenses are obviously very heavy, but are more compact than traditional mirror lenses. In spite of the innovative design, test results do not compare favourably with other mirror designs and refractor optics, so these lenses are now only curiosities. A 600mm f/8 and an 800mm f/11 were made (followed by a 500mm with a plastic aspheric front element and decidedly poor performance). There was even a 300mm f/5.6 the size of a normal (50mm) lens, but I have not been able to confirm whether this uses the same construction.