What exactly is an FBO and what services are provided? To most non-U.S. pilots who have little or no experience flying in the U.S., the concept of a FBO is not very well understood.

What exactly is a FBO and what are the services that it can provide? I've heard that they can refuel your plane, move it to a hangar, clean it, provide preflight planning facilities, etc.

Is there a charge for these services (I guess so) and if so, what can be expected? I've also seen many airports with multiple FBOs. How does that work?

All I know is that FBO stands for "Fixed Base Operator".

This may seem like a very stupid question but in Europe there's no such thing. If you need fuel you either taxi to the pump and fill up or call up the fuel provider (if you're lucky because they usually only serve private jets).
A Fixed Base Operator (FBO) is an entity - usually a commercial company - which provides "aviation services" at an airport. One of the main services they provide is fueling, either full service or a self service pump.  They don't all provide both 100LL and JetA, but most do.  

Larger ones provide other services, such as catering, lavatory service, pre-flight briefing computers and wifi access, ground power units, and tie down or hangar space for parking.  Charges vary from place to place, most large FBOs will charge a fee for aircraft parking, but sometimes waive a day or so of that if you purchase fuel.  They can often assist in arranging ground transport or hotels as well.

Most are open hours like 7AM - 8PM, but that varies as well.  Most self-service pumps can be used 24 hours a day.

Some have a "crew car" available to borrow for a short trip to get lunch or run an errand, though calling ahead to check on that is important.