In a reflecting telescope, the image is provided by a mirror rather than a lens. There are two advantages with this: there is only one surface that needs to be worked, unlike a lens which has two, and mirrors reflect light of all colours in the same way, so there is no false colour in the image, unlike refractor images. However, because the light is reflected back the way it came, the image is formed in the middle of the incoming light beam, so there is no room for the observer’s head. The solution to this was proposed in the 17th century by Isaac Newton, he of gravity fame, which was to use a small mirror to reflect the light sideways to the edge of the tube, where you can view the image with an eyepiece.
Such telescopes are still known as Newtonian reflectors, and the vast majority of reflecting telescopes work like this. The main mirror has a curved surface like a dish and is called the primary, and the small one the secondary or flat. It is held in the middle of the tube by an assembly of (usually) three or four vanes known as a spider.
Mirrors have a focal length just as refractors do. The curvature of the surface can have a spherical shape, and budget reflectors have this form, but better images are produced if the curve is modified slightly to be a parabola. The front surface of the mirror only is coated, unlike household mirrors where it is the back surface that is coated and a protective layer over the back is added to prevent scratching. But in astronomy a telescope mirror made in the same way, like a shaving mirror or compact mirror, would result in an unwanted reflection off the front surface of the glass. The front surface coating, however, is delicate and is susceptible to scratching even if dust is wiped away, so reflector mirrors need more care than refractor lenses.