There are two types of telescopes that one normally makes as a beginner. A simple refracting type, very similar to what Galileo Galilee used in 1610, and a reflecting type that was developed by Newton.
The refracting type uses a lens to collect light where as concave mirror is used for the purpose in reflecting type telescope.
Refracting type: The first of this kind of telescope was used by Galileo. It had a convex lens as objective and a concave lens was used as eyepiece. This optical configuration gave a rather small field of view.
It was later modified by Johannes Kepler who replaced the concave lens of the eyepiece by a convex lens. This configuration gave better field of view but the images are inverted or upside down, which is no consequence for astronomical observation.
This system suffered another problem, called chromatic aberration. When the light passes through a lens it breaks into its component colours and rays of different colours focus at different distances. Therefore the images are not quite sharp.
English optician John Dollond solved this problem by adding a concave lens after the convex lens of the objective. Such lens is called an achromatic lens.
During the International year of Astronomy in 1999, various institutions and clubs across the country conducted telescope making workshop of this kind of telescope. For example, IUCAA conducted a serious of workshop for making what they called the G++ telescope. This telescope had an achromatic lens and a 45 degree mirror. This was placed on simple stand. Using a telescope of this kind one can see the moons of Jupiter, phases of Venus, some bright nebulae, and even some galaxies.
Reflecting type: Isaac Newton made the first such telescope. This is also the most popular type among amateur astronomers. The process involves making a concave mirror and placing the mirror with the eye piece etc in a suitable mount.
Normally, amateur astronomers make concave mirror of roughly 150 mm (or 6 inch) in diameter. It takes about 3 to 4 weeks to make such a telescope. In the recent past, mirrors of 100 mm (4 inch) have become popular as these can be made in just about a week. Check this link Amateur Telescope Making - India.