Have you ever been waiting for your flight at an airport and noticed that big roll of paper being printed up at the counter? That is called the "Flight Release". That roll of paper has all the information that the pilots need to fly their flight. It has their fuel, their route, the weather information, probable payload, and everything else they need. A Flight Dispatcher put that together for them.
The pilot looks it over and calls the Dispatcher if there are any questions. The aircraft cannot push back until both the Captain and the Dispatcher agree that it is safe to do so.
Once the flight is airborne the Flight dispatcher keeps an eye on that flight and makes any recommendations to the Captain regarding any change in route, altitude, speed, destination or anything else relating to flight safety and the efficient operation of the flight. This is called "flight following". If ATC gives the Captain a route change, the Dispatcher will calculate the new fuel burn. If the flight has to hold, the dispatcher will calculate how long the flight can hold before it has to land somewhere.
The Captain and Dispatcher communicate with each other while the aircraft is in flight primarily by ACARS. Think of it as an instant messaging system between the Flight Deck and the Dispatcher's computer. They type messages back and forth to each other and each sees the message on a screen.
The Flight Dispatcher is also there to help if there is a problem with the flight, i.e., a sick passenger, a mechanical issue, a change in the weather, or any other problem that the Captain may be facing.
The Captain and Dispatcher have joint authority to initiate, delay, continue, deviate, or cancel a flight. Nothing can happen until they both agree on a plan of action. Because of this the FAA considers the dispatcher to be a "crew member" and the Dispatcher's base of knowledge is very close to that of the pilots. Because of this, Dispatchers are allowed to ride up in the Flight Deck if they want to and are required to do it at least five hours a year.
No, the Dispather is not an Air Traffic Controller. However, the Dispatcher's plan for the route is given to ATC. ATC then takes all the flight plans filed by all the Flight Dispatchers who work for the airlines and put them all together for the big picture and then make it happen.
In short, the Flight Dispatcher works for an airline to plan flights and give that information to both the flight crew and air traffic control, monitor the flights while they are airborne, and work with the Captain to solve any problems that may arise. Their main concerns are, in order, flight safety and flight efficiency.