There are many different types of atomic bombs, but the two main classes are Fission and Fusion. Fission creates energy by splitting atoms. Fusion creates energy by fusing atoms.
The "fission" atomic bomb works by setting off a chain reaction of decaying nuclei, which releases huge quantities of energy. Essentially, Uranium decay is induced when a free neutron collides with a uranium nucleus and causes it to become unstable and split, releasing two neutrons, heat, and gamma radiation. The cycle is then continued at a devastating rate.
I'm sure someone can explain a little more in depth, but I've got to run!
Oh, and see E=mc^2 for your second question :]