What, for example, is the difference between an atomic bomb and a hydrogen, or a thermonuclear, bomb?
Atomic bombs utilize nuclear fission to generate energy. Hydrogen bombs, also called thermonuclear bombs, utilize nuclear fusion. In fission, a nucleus of an atom of uranium or plutonium absorbs a stray neutron, becomes unstable, and splits apart, releasing energy and three more neutrons. These neutrons are absorbed by more nuclei causing more splitting (or fissioning) releasing more energy and more neutrons and within a fraction of a second there's a chain reaction in which all the atoms of the fuel are split. A mass of plutonium the size of grapefruit can produce the energy equivalent of 13,000 tons (13 kilotons) of TNT. This was the approximate yield of the bomb dropped on Hiroshima in 1945.
Hydrogen bombs are even more powerful. The average hydrogen bomb has a yield of about 1 million tons (one megaton) of TNT, but some can produce as much as 50 megatons. This is about 4000 times more powerful than the Hiroshima blast.
These weapons rely on the energy released in the process of nuclear fusion. When hydrogen nuclei are forced together under great pressure they fuse and convert a small amount of mass into energy according to the equation E=MC^2. The pressure is generated by first producing a fission explosion. This generates enormous heat which in turn generates enormous pressure which forces the hydrogen nuclei together producing a fusion explosion. Thus, a hydrogen bomb is actually a combination of a fission and a fusion explosion. Because these weapons are triggered by the intense heat produced in a fission chain reaction they're called "thermonuclear" weapons.
This is video of a 1954 test of a 15 megaton weapon:
It's not clear whether the North Koreans have thermonuclear weapons or only fission weapons (which are bad enough). What most nuclear nations do is mount these weapons on missiles (called nuclear missile warheads) as well as drop them from planes as bombs, and this capability is what both North Korea and Iran are trying to achieve. If they're successful it'll have a seriously destabilizing effect on the world as all of their neighbors seek to develop their own arsenals to counter those of Iran and North Korea.
The enormous blast power of a single one megaton warhead can destroy an entire city. The radioactive fallout from the blast could kill tens of thousands more downwind from the blast. Such weapons in the hands of fanatics and madmen is a frightening prospect which is why so much of the world insists that neither Iran nor North Korea must be allowed to develop the capability to mount these weapons on missiles.
Unfortunately, it may take war to prevent them from obtaining those nuclear missiles.