The classic amateur telescope is a Newtonian reflector with purchased eyepieces.
The Newtonian reflector has only two optical surfaces: the primary mirror, and a small flat mirror which is often purchased. This makes it relatively easy to construct. The mirror focuses light back upwards through the tube to a secondary mirror, usually an optical flat. The flat reflects the light sideways to an eyepiece mounted on the side of the telescope. The image is inverted from what the eye would see.
The mirror is usually rough ground to a shallow spherical section, and then gently "figured" to a parabola using a special polishing lap.
The construction proceeds in four phases.
First, the amateur decides what size to construct. The difficulty of construction grows roughly as the square of the diameter of the mirror. A 4-inch mirror is a moderately easy science-fair project. An 8 inch mirror is a good compromise between ease and constructing an instrument that would be expensive to purchase. A 12 inch mirror is difficult, and a telescope over 24 inches usually must be ground and lapped with mechanical assistance. Amateurs have constructed telescopes as large as a meter across (39 inches), but this is foolhardy for anyone other than the best-funded, experienced clubs.
Mirrors are usually ground from low-expansion borosilicate glass (Pyrex (TM) is the brand name). Alternatively, a special ceramic called Cer-vit produces superior mirrors, but costs more.
The mirror blank is ground against a blank made from thick window glass. The amateur moves the mirror blank back and forth, periodically stepping sideways to rotate around the work-table to average motion errors. Periodically, one sprinkles water and fresh abrasives on the blank.
After a bit, the mirror's curve is deep enough, and one begins to fine-grind the mirror. The same basic step is repeated, using successively finer grits, until one reaches 800 grit (fits through an 800-wire-per-inch mesh).
It is heartbreaking to partially complete a fine-grind, and then find a scratch from a larger size of abrasive. Clean the system carefully when reducing grit sizes.
After that, one constructs a "lap" from tar or a mixture of rosin and beeswax. The lap has channels cut in it to let water and abrasives run off.
Then, using the lap, one begins to polish the mirror using rouge. The scratches of the rouge are smaller than a wavelength of light, and the mirror thus becomes a specular (mirror-like) reflector.
At some point, the mirror is polished, but it is still a section of a sphere. At this point, a light and knife-edge are set up at the focus of the mirror. Using this arrangement, viewing the mirror past the edge will show shadows showing the "figure" of the mirror.
The lap is cut away, or a different stroke is used to polish more in the center. When the above inspection method shows a slight circularly-symmetric doughnut appearance, then this indicates the sought-for parabolic mirror shape.
The mirror is then aluminized, by placing it in a vacuum tank with electrically-heated nichrome coils that can evaporate aluminum.
The mirror is then mounted in a mechanical tube. This is often the most difficult stage. However, it also requires no esoteric skills.
The classic reference is a set of books called "Amateur Telescope Making."