Walk into a telescope shop and the instruments look like three unrelated products, which is roughly correct. This page is how to identify each design on sight and what each one is trying to be good at.
How to tell them apart on sight
If the eyepiece is at the back and the tube is sealed with a lens at the front, it is a refractor. If the eyepiece is in the side near the top and the tube is open, it is a newtonian reflector, and if it sits on a plywood box it is a dobsonian. If the tube is short and fat with a lens at the front and the eyepiece at the back, it is a catadioptric.
What each is trying to do
The refractor is trying to give the cleanest possible image per millimetre and never need maintenance. The reflector is trying to give the most aperture for the money. The catadioptric is trying to fold a long focal length into a portable package that a mount can track. All three succeed, at different prices.
Reflecting against refracting, in one line
A reflector bounces light off a mirror and a refractor bends it through a lens. The mirror is cheaper to make large and needs occasional alignment; the lens is sealed and maintenance free and expensive at any size. That single difference explains almost every other difference between them.
Which to buy, in one line each
For the most to see at the price, a dobsonian. For grab and go with no maintenance, a small refractor. For portable tracking and planets, a maksutov or schmidt cassegrain. For photography, a fast apochromatic refractor on the best mount you can afford.
Questions people ask about telescope
What are the different types of telescopes?
Refractors with a lens, reflectors with a mirror, and catadioptrics using both. Every amateur instrument is one of those three.
What is the difference between reflecting and refracting telescopes?
A reflector uses a mirror and gives more aperture per pound with occasional collimation; a refractor uses a lens, is sealed and maintenance free, and costs far more per millimetre.
Which telescope design is best?
None overall. Reflectors win on value, refractors on convenience and photographic star quality, catadioptrics on portability with tracking.