"Fair use" is one of those nebulous concepts that gets thrown around a lot without people really understanding it. While it's true that it can give some creative leeway in parodies and reviews, it's not a one-size-fits-all solution to creating fan content. In fact, it's entirely possible for the owners of intellectual properties to crack down on fanart if they so choose.
As it turns out, though, it's very rarely in the interest of those IP owners to do so. Fanart brings in very little revenue to the fanartists while it simultaneously fosters the fandom's community, and the potential backlash from shutting down fans is usually not worth it. Neil Gaiman has an excellent video on piracy that captures the spirit of this very well.
Even so, I do recommend taking precautions when you make fanart, just to be safe. Avoid including character names or series logos. NEVER try to sell anything that was traced or directly copied from an existing piece of art. And finally, always sign and include a copyright symbol and date on your fanart, because even if you didn't create the original characters, YOU OWN THAT PIECE OF FANART.
Rock on, guys, and never stop drawing.