Historically, blotting paper or cloth were used to extract filter coffee.
Modern coffee filters of paper are made from about 100 g/m crêped paper. The crêping allows the coffee to flow freely between the filter and the filtration funnel. The raw materials (pulp) for the filter paper are coarse long fiber, often from fast growing trees. For example, Melitta uses up to 60% of bambus in their filters since 1998. Both bleached and unbleached qualities are made. Coffee filters are made in different shapes and sizes to fit into different holders.
Most notable are the (paper) coffee filter systems introduced by Melitta (1908, 1932, 1936, 1965), Chemex (1941) and Hario (2004). 
Important parameters are strength, compatibility, efficiency and capacity.
Tea bags also work as a kind of paper filter. They are made from abacá fibers, a very thin and long fiber manilla hemp. Often the paper is augmented with a minor portion of synthetic fibers. The bag paper is very porous and thin and has high wet strength.