There are various distinct types of telescopes. But the most easily recognized as a classic telescope is the refractor (also known as a Newtonian) type. Others are Dobsonians, and Smith-Cassegrains, which are more compact in overall lenght.
The word "equatorial" refers to the fork-like mounting base which goes on top of a tripod. Nowadays these mounts are motorized and computer controlled to automatically point the scope to any chosen position in the sky.

Regards.
Luis Luis