Primarily, nap is the raised (fuzzy) surface on certain kinds of cloth, such as velvet. Nap can refer additionally to other surfaces that look like the surface of a napped cloth, such as the surface of a felt or beaver hat.
Starting around the 14th century, the word referred originally to the roughness of woven cloth before it was sheared. When cloth, especially woollen cloth, is woven, the surface of the cloth is not smooth, and this roughness is the nap. Generally the cloth is then 'sheared' to create an even surface, and the nap is thus removed.