A hydrogen bomb is a “two-stage weapon”, in which an atomic bomb is the first stage.
An atomic bomb works by using conventional explosives to super-compress a core of super-heavy radioactive metal, like uranium or plutonium. In this dense state, this super-heavy metal, the “nuclear fuel”, reaches a state where the neutrons the fuel naturally loses over time have a high chance of hitting other nuclei in the fuel, causing those nuclei to split, releasing more neutrons that hit other nuclei, and so on and so forth.
This reaction generates a lot of energy, which increases the temperature in the center of this core, for an instant, well beyond the temperature of the sun. That high temperature, and the high density of the core, is what is used in a hydrogen bomb.
In a hydrogen bomb, the conventional explosives compact the core, as before, but now that core is hollow. Instead of metal in the center, now you have a lot of a type of hydrogen, called tritium. This isotope of hydrogen has two neutrons along with the proton in the nucleus. This hydrogen gets super-compacted and super-heated in the center of the fission reaction, until the overall energy level gets high enough to allow for fusion — the hydrogen atoms become a helium atom. Two neutrons from the fusing tritium atoms are released, along with even more energy than we get from the nuclear fission, and this “boosts” the fission reaction by providing more neutrons for more fissions which in turn keeps increasing the temperature of the reaction’s core, in turn allowing higher rates of fusion.
All of this, in both cases, happens in a small fraction of a second, but the end result of a hydrogen bomb is a dramatically higher energy output from the nuclear fusion at the very center of the reaction, up to a thousand times the explosive yield for a device of the same size.