What is the difference between a hydrogen bomb and an atomic bomb?

Atom bombs work by the principle of atomic fission (splitting large atomic nuclei), while hydrogen bombs work by atomic fusion (combining small atomic nuclei). The hydrogen bomb is hundreds or thousands of times more powerful than the atom bomb. The hydrogen bomb uses an atom bomb as a trigger.

The term "atomic bomb" is a general term that can be applied to any nuclear weapon. What kind of weapons are there and where does the hydrogen bomb fit in?

There are fission devices (the "regular" atomic bomb), fission-fusion devices (the clean hydrogen bomb) and fission-fusion-fission devices (the dirty hydrogen bomb).

In the atomic bomb (fission device), uranium or plutonium is forced into a "critical mass", causing the atoms of the element to fission or "split" into the smaller atoms of other elements. When they split, they give off neutrons that split even more of the atoms (i.e. chain reaction). Each atom gives off a tremendous amount of energy as a tiny fraction of its matter is converted.

In the clean hydrogen bomb (fission-fusion device), the heat given off by a fission explosion is directed at a container of fusible hydrogen (deuterium and/or tritium). The heat and pressure causes the hydrogen to fuse into helium, the same process that takes place in the Sun and stars. This reaction produces an incredible amount of energy, because again a tiny amount of matter from each atom is converted.

In the dirty hydrogen bomb (fission-fusion-fission device), the energetic neutrons from the fusion explosion are so numerous that a casing of "ordinary" uranium (mostly U-238) will also fission, creating a fantastic amount of energy (up to 90% of the total yield of the bomb can be from this fission). Thicker casings or additional stages could theoretically create massive bombs 1000 times the power of fission bombs. The largest bomb ever tested, the 50-megaton "Tsar Bomba" of the Soviet Union, was built with this design (three stage design: fission primary, fusion secondary, fusion tertiary). If it had used actual uranium around the third stage, it could have yielded 100 megatons or more. However, the fallout from such a bomb would be large and widespread, risking contamination of areas far beyond the target. In the configuration tested, the "Tsar Bomba" was actually the cleanest nuclear bomb ever detonated (in terms of amount of fallout per kiloton of yield), even though it produced more total fallout than any other nuclear bomb ever detonated (because of the very high yield).

The design used by modern weapons was created by the physicists Edward Teller and Stanislaw Ulam in 1951.
The "Hydrogen" bomb refers to the "Fusion" of a Hydrogen Isotope on an Atomic scale by way of steps of multiple reactions thus yielding a much more powerful explosion upwards of 500 Million Tons of TNT. It is also known as "ThermoNuclear". The "Atom" or "A" bomb refers to the "Fission" or "Fusion" of Uranium or Plutonium in a single step reaction, rather than multiple steps,yielding an explosion.