system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: The differences between atomic, nuclear, and hydrogen bombs.
gpt
Short version: "Atomic bombs" use nuclear fission (the splitting of atoms) to make a pretty big explosion. "Hydrogen bombs" use nuclear fission _and_ nuclear fusion (the combining of atoms), and in theory can make explosions without any upper limit. "Nuclear weapons" can mean either. 

Elaboration:

"Atomic bombs" derive their energy exclusively from the fissioning of uranium-235 or plutonium-239. The bomb dropped on Hiroshima, for example, involving one piece of uranium-235 being shot into another with conventional explosives. This created a supercritical mass, starting a nuclear fission chain reaction. The bomb dropped on Nagasaki was of a different design, one where specially-created explosive "lenses" of a TNT-like explosive were used to spherically compress a ball of plutonium to very high density. This too created the supercritical mass and subsequent explosion. The latter type of bomb, the implosion bomb, is the most common type of atomic bomb. The largest pure-fission bomb ever created by the USA was around 500,000 tons of TNT equivalent. (The Nagasaki bomb, by comparison, was about 20,000 tons of TNT equivalent.) There is an upper-limit on the yield due to the fact that having too much fissile material (uranium-235 or plutonium) in one place is dangerous.

Hydrogen bombs derive their energy from both fission and nuclear fusion. Basically they work like this: an atomic bomb (of the implosion type) sits at one end of a heavy container. At the other end is a lump of fusion fuel (a material made of very light atoms, usually an isotope of hydrogen or lithium). At the center of the fusion fuel is another lump of fission fuel, and the fusion fuel lump is wrapped in a heavy material (e.g. depleted uranium or lead). When the atomic bomb inside the casing explodes, it first releases a burst of X-rays and gamma rays. These are reflected off of the inside of the casing, and end up compressing the fusion fuel. As it compresses, it starts another fission reaction at its center. Pushed from both the outside and the inside by essentially two atomic bombs, the fusion fuel starts to undergo nuclear fusion. This releases a lot of energy, and a lot of neutrons. These neutrons can then induce fusion in the casing around the fusion fuel, if it is fissionable material (like depleted uranium, which normally doesn't fission, but under such conditions can). The result can be a very large explosion — many millions of tons of TNT equivalent. In most modern hydrogen bombs, about half of the yield comes from fission, and half comes from fusion.  All of the weapons in the current US arsenal are hydrogen bombs developed to fit into a very small space, so the warheads are only about the size of a trash can but they have an explosive yield of 200,000-400,000 tons of TNT. In theory, you can make hydrogen bombs as big as you want, because fusion fuel has no maximum limit (unlike fission fuel), and you can "chain" such bombs together, one bomb setting off the next. The only limitation is practical size. The largest hydrogen bomb ever detonated had a yield of 50,000,000 tons of TNT equivalent, and was about the size of a school bus. 

"Nuclear weapons" is a category for the above that doesn't really distinguish between the above two major types. It became more popular in an era (the late 1950s onward) where people stopped really caring about whether fission or fusion was present in the warhead or not. It is, strictly speaking, a more accurate team, as "atomic" is not really specific (electrons are part of atoms, too, so chemical reactions are just as "atomic"; it is the nucleus that makes nuclear reactions stand out). The term "thermonuclear" is used when it is intended to indicate that fusion reactions are taking place, without using the "hydrogen bomb" notation.