A bomb: atomic bomb (fission)
H bomb: Hydrogen bomb (fusion)
Fission: a large atom (for example, Uranium or Plutonium) splits into smaller atoms. It takes a lot less "binding" energy to hold together small atoms than it takes to keep a large atom together. It is the difference that is liberated as energy; slowly in a reactor (to produce electricity) or very rapidly (bomb). The break-even atom is iron. To split something smaller than iron, you have to add energy. To split a larger atom, you can liberate energy.
Fission can spontaneous. Or it can be triggered by having a neutron hit the atom. It just so happens that fission also liberates neutrons. If you have enough fissile material (the sufficient mass is called "critical mass"), then the liberated neutrons will always hit other atoms which will, in turn, liberate more neutrons as they split and these new neutrons will hit other atoms... a chain reaction. For Plutonium, the critical mass is more than 6 kg.
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Fusion: small atoms (or atomic particles) fusing together to form larger atoms. As long as you work with atoms much smaller than iron, you will liberate energy as it takes a bit less "binding energy" to hold a larger atom than a small one.
The reaction with the best "profit" is going from 4 Hydrogen to 1 He (it takes many steps -- check out fusion on Wikipedia). However, the natural repulsion between the protons (they are all positive, therefore they push each other away) means that you need energy to start the process. In the Sun, this is done with very large pressure and temperatures near 15 million degrees. The protons are so close together (pressure) and hit each other so fast (temperature) that they overcome the repulsion.
The energy released by the fusion is more than the energy needed to trigger it.
For a bomb, the energy required to fuse the light atoms in the H-bomb is given by an A-bomb surrounding the core.
The A-bomb is triggered to explode with a lot of energy directed inwards (a lot still spills outwards), which fuses the light atoms and triggers the H-bomb.
microseconds of fun for all.