Reflecting telescopes
A reflecting telescope is one that uses curved mirrors to focus the image of the stars or planets. These can be built much more compactly than refracting telescopes.
Most large telescopes in use today are reflecting telescopes. It is easier to build a very large mirror than a very large lens, in part because only one side of the mirror is important. (A lens must be supported at its edges only, which is difficult technologically.)
Newtonian telescope
The Newtonian telescope uses a large curved mirror to focus the light, and a small flat mirror to divert it to the eyepiece.
This design, in use since the late seventeenth century, is one of the simplest ways of building a reflecting telescope, and is still commonly used by amateurs.
Cassegrain telescope
The Cassegrain design uses two curved mirrors. The large mirror focuses the light in the same way as in a Newtonian telescope, but has a hole in the centre. The second mirror reflects the light back straight through the centre of the larger, to the eyepiece.
The Cassegrain telescope was developed only shortly after Newton's reflector, by Laurent Cassegrain. This design helps to protect the primary (larger) mirror, which is one reason why it is much more popular among professional astronomers than the Newtonian design, despite being more difficult to build.