Atomic bombs only use nuclear fission, splitting the nucleus of an atom, splitting plutonium and/or uranium into smaller atoms in a chain reaction which releases massive amounts of energy.
The bombs dropped by the US on Hiroshima and Nagasaki during WWII were both atomic bombs.
A hydrogen bomb, also called a thermonuclear bomb or an H-bomb, uses a second stage of reactions to magnify the force of an atomic explosion.