Refracting telescopes
A refracting telescope is one that uses lenses to focus the light from the stars or planets being observed. The light passes through the lenses going the same direction the whole time, before it reaches the eye of the observer. The shape of the lenses determines how well the telescope works.
However, the more effective the refracting telescope, the bigger it generally needs to be. Thus, this type of telescope is rarely used for any big astronomical projects.
Galilean telescope
The telescope that Galileo used to make his astronomical observations is commonly known as a Galilean Telescope, though he did not invent it.
It is composed of a convex objective lens and a concave eyepiece lens.
A telescope of this sort provides an image which is the right way up. However, it also has a very small field of view, which means that only a small part of the sky can be seen at any one time, and the image will be a little bit distorted because of the shapes of the lenses.
Keplerian Telescope
Johannes Kepler improved the design of the telescope by using two convex lenses rather than a combination of concave and convex.
This type of telescope provides a much wider field of view than Galileo's telescope. More can be seen at one time. The Keplerian telescope also inverts the image. This was inconvenient for terrestrial observation, but astronomers soon became accustomed to it.