We can think of reflector telescopes are those that use mirrors for gathering light rather than rather that lenses (refractors). They were first conceived soon after the development of the refractor in the 1600’s but largely due to the poor quality of mirrors available back then did not become popular until the mid 1700’s. While there were many scientists involved in the conception of the reflector, we generally think of Isaac Newton as the first to develop a working reflector and indeed to this day they are often called Newtonian telescopes.
Reflector telescopes are popular for several reasons, not least of which is that they allow for quite large diameter apertures to be achieved more practically and at lesser expense than refractors. Indeed most of the large telescopes used for professional astronomy are reflectors, even that most famous telescope, the Hubble! In astronomy the larger the aperture, the more light gathered and therefore the better the image and magnification that can be achieved.
Reflectors also suffer less from chromatic and spherical aberration due to the use of finely made parabolic mirrors so tend to produce good images for the money invested. They do require a little maintenance, particularly a process known as collimation but once you have done this once or twice it becomes a minor task you can easily take care of without much thought or effort.
Reflectors offer a great balance of aperture and price as their mirror construction is cheaper than the lens construction of the equivalent size refractors. For this reason they are very popular with amateur astronomers. Larger reflectors however do tend to catch all the light pollution of the city and suburbs as well as the interesting stuff so they are better suited to dark sky locations away from the lights.