All in all there are ##6## kinds of telescopes.
Optical telescopes is divided into two the refracting telescope and the reflecting telescope. This are like observatories in Universities.
http://www.wired.co.uk/news/archive/2013-03/05/uk-fund-optical-telescope
Refracting telescopes uses series of lenses in viewing. http://www.davidreneke.com/telescopes-for-sale/
(Look at the reflecting and refracting telescopes they almost look alike but the difference is that the refracting telescope has an eyepiece.)
Reflecting telescopes uses series of mirrors. There is a special kind of this telescope Multiple Mirror Telescope which uses many small mirrors rather one large mirror.http://www.davidreneke.com/telescopes-for-sale/
Infrared telescopes are telescopes which detects invisible energy. Such invisible energy are the infrared rays or sometimes known as heat energy.
X-ray telescopes are telescopes that detects x-rays given off by dying stars and others. They are outside because some of the rays are block by our atmosphere. http://binoculas.net/telescope-x-ray/
Radio telescopes detects radio waves (another form of radiation). This kind of telescope is the most familiar of all telescope. This looks like this:http://www.unawe.org/updates/unawe-update-1147/