Refracting telescopes use a lens as their primary optical element. When the light rays from a star hit the primary convex lens, they converge or focus to a point – called the focal point. Placing the eyepiece at this focal point will magnify and focus the image. The larger the diameter of the primary lens, the better resolving and light gathering power the telescope will have.
For a given diameter, high quality refractors produce better image quality than reflectors, but there is a price. A high quality lens must be optically perfect on the front surface, interior, and back surface. A mirror on the other hand only needs to be optically perfect on the front surface. Therefore refractors are more expensive than reflecting telescopes of the same diameter. There are also many small cheap refracting telescopes on the market with poor optical quality lenses.
Reflecting telescopes use a mirror as their primary optical element. Light rays reflected off a concave mirror will converge or focus to a focal point. Placing the eyepiece, eye, and the attached rest of the astronomer at this point will cause the obvious problem of blocking the light from the star being observed. However for extremely large research telescopes, four or more meters in diameter, only a small fraction of the total light is blocked. On these large research telescopes, this configuration can be used and is called the prime focus.
For smaller telescopes a way must be found to place the eyepiece and observer out of the incoming light path of the telescope. Many ways have been found to do this, but the most common are the Newtonian and Cassegrain designs.
The Newtonian reflecting telescope, devised by Isaac Newton, uses a small flat secondary mirror placed just before the focal point. This secondary mirror is tilted at a 45 degree angle so the eyepiece can be placed outside the incoming light path at the top of the tube. Newtonian reflectors have the advantage of the lowest cost for a given diameter and quality. They can however be bulky. While having the eyepiece at the top is no problem for small telescopes, it is inconvenient for larger instruments.
Most large astronomical telescopes are therefore Cassegrain reflectors. A Cassegrain reflector uses a small convex secondary mirror placed before the focal point. The secondary mirror reflects the light through a small hole in the center of the primary mirror. The focal point and eyepiece can then be at the bottom of the telescope, which is more easily reached in the case of a large telescope, read more from here.
Catadioptric telescopes, such as Schmidt Cassegrain telescopes, employ both a lens and mirror. In simple terms they are Cassegrain reflectors with an additional lens, often called a correcting plate, placed in the light path before the light strikes the primary mirror. The correcting plate improves the image quality, so these telescopes have very good quality images. They are also more compact that Newtonian reflectors of the same diameter. They cost more than Newtonians but less than refractors of the same diameter and quality.