An FBO is a Fixed Base Operatior. This is a business that caters to general aviation (including business aviation). FBO's have a wide range of services ranging from just providing fuel, to providing flight training for all ratings, lounges, bunks, showers, and various other amenities(sp?).
When you fly your own general aviation airplane to another airport (that is staffed) you generally park at an FBO. A typical FBO would probably offer fuel, tiedowns, snacks, flight planning, often a pilot shop, and some can hook you up with a rental car. A few also have courtesy cars availble (cars that you don't have to pay for).
Some FBO's also offer flight training, and if you aren't taking your flight training through a school, you are probably doing it from an FBO.
Hope this helps