When light hits a surface that it can't travel through, it bounces back. If the surface is smooth, like a mirror, the light will reflect in a predictable way. If the surface is flat, the angle at which a beam of light approaches the mirror will be equal to the angle at which the beam is reflected, so i = r in the diagram below.
Curved mirrors can bend light and make parallel light rays converge to a focus. This focus is directly in the path of the incoming light, so there are several ways of making images from the mirror visible. One is called a Newtonian reflector, where a flat mirror is used to point the light rays out to an eyepiece.
There are several other types of reflectors that solve the issue of where to focus the light in different ways. Cassegrain reflectors have a convex secondary mirror and a hole in the middle of the primary mirror. Prime focus telescopes have no secondary optics and the observer or camera observes the image from near the focal point. Coudé telescopes use a convex secondary mirror like a Cassegrain and an angled mirror like a Newtonian reflector to move the light rays to a focal point away from the telescope. This arrangement is useful when optical equipment is being used that is too heavy to mount directly on the telescope.
Reflecting telescopes have many advantages over refracting telescopes. Mirrors don't cause chromatic aberration and they are easier and cheaper to build large. The are also easier to mount because the back of the mirror can be used to attach to the mount. Reflecting telescopes have a few disadvantages as well. Because they are normally open, the mirrors have to be cleaned. Also, unless the mirrors and other optics are kept at the same temperature as the outside air, there will be air currents inside the telescope that will cause images to be fuzzy.
Different reflectors use different shapes of mirrors. Parabolic mirrors will focus all incoming light rays to a single point. However, images from a parabolic mirror will have a defect called coma, where images far from the center of the field of view are elongated. A spherical mirror surface is relatively easy to make, but different parts of a spherical mirror have slightly different focal lengths, so images will be fuzzy. Mirrors in modern telescopes area made in various shapes to correct for these errors. Some telescopes use a combination of mirrors and lenses. Schmidt-Cassegrain telescopes use a spherical mirror with a correcting plate that corrects the focus.
LCOGT's 1.0 meter telescopes are quasi-Ritchey-Chrétien telescopes. A true Ritchey-Chrétien has a hyperbolic primary and a hyperbolic secondary mirror. In the design of LCOGT's 1.0 meter telescopes, the shape of the mirrors has been changed a bit in order to find a more optimal optical design for the system as a whole. Because the mirror shapes have changed, the 2 mirrors alone no longer are a Ritchey-Chrétien telescope in the strict definition of the design. It is still close though, hence the name, "quasi-RC."
This animation shows how light travels in LCOGT's 2.0 meter Faulkes Telescopes.