The
hydrogen bomb, also known as a
staged thermonuclear weapon, is indeed the successor to the atomic bomb as
CrazyIvan has noded above. It is not the only weapon type that uses fusion reactions, however; see the
boosted fission weapon for more. The hydrogen bomb is also known as a
fission-fusion-fission weapon in some instances due to its design.
How does it work?
Glad you asked. I'll try to expand on what's already been offered here. The first thing you need to know about a fusion weapon of any kind is that it has at its heart a plain ol' fission weapon, aka atomic bomb. This is usually called the initiator or trigger. This is because fusion reactions are incredibly hard to start, and require enormous amounts of energy to coax them into existence. The only way of doing that in the field, as it were, without buildings full of energy storage and devices, is to use an atomic bomb.
So at the core of the fusion device is a fission device. There is a neutron reflector as well, just like in a boosted-fission weapon, which ensures that the energy derived from the atomic explosion of the trigger is properly utilized and focused on the fusion fuel. That fuel is typically one or both of two types of substance. The first is tritium, which is an unstable gaseous isotope of hydrogen. This substance, while easier to fuse, is short-lived (half-life of ~12 years) and hard to keep on the shelf, being gaseous and radioactive. So while there is usually some tritium inside these weapons to help start things along, it's not the primary fuel so that it can be replaced when necessary and so that a more efficient means of storage can be used.
There are three types of hydrogen fusion that are relevant here; they are T-T fusion, D-T fusion, and D-D fusion. The T is tritium and the D is deuterium, another isotope which is more stable. Although that makes it easier to store, it also makes it harder to fuse, requiring the kick of tritium via a T-T or D-T reaction, or a very very efficient fission trigger to set it off. Deuterium is usually present in modern thermonuclear weapons as lithium deuteride, a solid compound which can be safely and easily stored, shaped and transported.
The next problem is how to focus enough energy on the fusion fuel to ensure that it fuses. There are at least two popular methods; the first is to place the initial fusion fuel (usually the tritium) inside the fission core. This way it receives the full implosive force of the fission blast. The lithium deuteride is then placed around the device, inside a neutron reflector and tamper (tamper because its inertia 'tamps' the blast for a few micro- or milliseconds, usually made of U-238). The problem is that by the time the tritium fuses, the fission explosion has already begun moving the tamper out of the way. Also, replacing the unstable tritium is hard if it's in the center of the bomb.
A more efficient and maintenance-friendly way to do this is to use a neutron reflector that's in the shape of an ovoid or egg. An ovoid has two geometric foci within it. Any ray drawn outward from one focus will pass through the other after reflecting off the outer shell of the ovoid. Thus, if you place the fission trigger at one focus and the fusion package at the other, all the energy of the initial blast that is reflected by the neutron reflector will be concentrated on the fusion fuel, which being separated from the trigger will enjoy a few more microseconds of existence as the neutrons are faster than the blast wave coming from the trigger. Thus inundated with energy, the fusion package will fuse, and the fusion reaction will then release far more energy than the fission reaction alone. Plus, the weapon can derive most of its energy from fairly stable compounds (lithium deuteride) which aren't very harmful and are cheaper to get. Deuterium is present as a small but not invisible percentage of the hydrogen in the world's oceans, as one or both of the 'hydrogen' atoms in a molecule of water. Such water is known as heavy water, and has several uses in the fission/fusion industries. Finally, the tritium in the fusion package (if any) can be replaced without touching the fission triggering device.
This method of utilizing only the radiation pressure inside the weapon, however, is not the most efficient. Although the precise method of achieving fusion ignition inside the Teller-Ulam devices that make up modern weapons is still highly classified, some open-source analysts have come to the conclusion that the most likely means is known as tamper ablation. In this model of what goes on, the initial detonation of the primary is surrounded by a shell (typically the weapon casing) known as the hohlraum ('radiation room', literally). The energy from the first detonation of the primary is reflected or absorbed by the inner layer of the hohlraum in the form of X-rays. This causes the inner surface to turn to plasma and begin to compress the weapon's contents inwards. The secondary fuel (lithium deuteride) is placed within the hohlraum inside a shell of either U-238 or lead. The outer surface of this shell (the 'tamper') ablates as well - and the reaction from the ablation outwards presses the contents of the tamper inward. This heats and compresses the fusion fuel within, which eventually causes fusion. There may be a rod of fissile material at the core of the fusion fuel such that when it compresses far enough, it causes critical mass in that material, and is hence squeezed between two pressure waves.
The final step of the game is the fission-fusion-fission weapon. This device is engineered so that the resultant fusion reaction causes the normally quite stable U-238 of the tamper to fission as well, as the energy liberated by the fusion explosion is enough to start that reaction. This will tend to toss off a lot more fallout, as the U-238 will not all be consumed, and a great deal of it will be atomized and tossed into the surrounding environment. Of course, it is a more efficient way of utilizing the components of your fusion weapon. This can be avoided by using lead or another non-fissile material for the tamper, albeit perhaps at a cost in efficiency.