A full stack web developer is one that deals with both the back-end (the server side) and the front-end (the client side, the browsers).
The whole set is called a "stack" because it's a "stack" of technologies (think server technologies on the bottom and client technologies on top). So "full-stack" is that developer who deals with all layers of the stack, not just a part of them.
There's an article on SitePoint where the author talks about what it's like to be a full stack developer today, including listing many of the technologies you may have to deal with. It includes the following illustration of the development layers: