Telescopes work by collecting a lot of light and then use mirrors (Reflecting telescopes) or lenses (Refracting telescopes) to focus that light, i.e. to bring it together at a point. We can use ray diagrams to show how the light goes through the lenses and reflects off the mirrors inside the telescope.
Here are some ray diagrams for a number of different telescopes:
The example ray diagram above shows three imaginary "rays" of light being reflected off a mirror and then going through a lens. You can see where the focus is by looking where the "rays" cross each other. It is the lens that is making the light focus. If you want to use a mirror to focus light, then it needs to have a curved surface. It is these curved mirrors that are used in telescopes.
The light enters the telescope from the left and is focused by the main Lens. The Eyepiece (another, smaller lens) is used to magnify the image for us to look at.
The light enters the telescope from the left and is focused by reflecting off the large Concave Mirror. Since the focus is inside the telescope, it is difficult to see the result or put an instrument there. Normally telescopes use another mirror to move the focus somewhere else (see the ray diagrams for Newtonian and Cassegrain telescopes).
The light enters the telescope from the left and is focused by reflecting off the large, curved Primary Mirror. A flat mirror is then used move the focus outside the telescope where it is easier to see, or to put an instrument to measure the light.
The light enters the telescope from the left and is focused by reflecting off the large, concave (curves inwards) Primary Mirror. A convex (curves or bulges outwards) secondary mirror is then used to move the focus right down through a hole in the centre of the primary mirror, where the instruments are located.
Most modern professional telescopes are of this type, including the Liverpool Telescope.