A fan artist (or fanartist — in line with fanwriter and faneditor) is both a fan and an artist. Fanartists do their art for the benefit of other fans without more than nominal payment. (The key point is whether the payment is large enough to form a significant part of the motivation for creating the art. A beer or copies of a fanzine or $10 is not enough. And undying friendship may have value beyond mere money, but it's not a financial motivation.)
The subject of the art can be anything at all. Most fan art is not illustration as such. Although some faneds do ask for them, pieces illustrating an article or a specific character are a minority of fan art. Cartoons are popular, as are stfnal themes. Until the advent of inexpensive color printing and online art, most fan artists worked in black and white, doing the line drawings that reproduced best for mimeography, particularly in the days of handstencilling.
A professional artist who contributes art to a fanzine or convention program book without being paid has done fan art. (This is more common than most people think.) Likewise, if an artist who is well-known for contributing fan art to fanzines does a piece of art in the same style and with the same subject matter and sells it for nontrivial pay, then that art is professional.