Filters can be glass, plastic, or acrylic, in rounded, thread-in shapes, or in square or rectangular slide-in filters that mount in slots, or uncommon these days, made of very thin gelatin material which can be held in a holder or even taped to the lens! Other even less-common filter mounting types types use bayonet-on mounting systems, and another older type is a rounded, metal-rimmed filter that has no threads, but which is held in position in a receiving ring, and then captured and held in place by a thread-in retaining ring.
The most-useful filters are the polarizing filter, graduated neutral density filter, neutral density filter, variable neutral density filter, and the various types of diffusion filters, such as the Tiffen Black Net Diffuser series. There are also some useful soft focus filters, like the Zeiss Softars.
UV filters are commonly used, and not all that useful withy digital cameras. UV and clear 'protective filters' can be bought by dealers at low wholesale prices, and then marked up obscene percentabes, and sold for tidy profits, using the "fear" approach to salesmanship. The more-costly a lens, the more-willing many customers are to pay $119 for a $5.59 wholesale priced filter.