Atomic bombs work by splitting atoms, while hydrogen bombs, also known as nuclear bombs, draw their energy from fusing atoms together. While both types of reactions involve the nucleus of the atom, and both can therefore rightly be called "nuclear," tradition has reserved the word for the types of reactions that are exclusive to fusion devices.
An atomic bomb works by splitting the atomic nuclei of heavy elements to produce atoms of daughter elements along with a large quantity of energy. It is this energy that provides the atomic bomb with its devastating force. Nuclear or thermonuclear weapons, on the other hand, work by heating up atoms of a light element, typically hydrogen, to the point that collisions between them result in fusion into a single atom of helium. This reaction also releases vast quantities of energy, which is the source of the nuclear bomb's destructive potential. Nuclear bombs use standard, fission-type atomic bombs as their explosive trigger to generate the necessary heat and pressure for nuclear fusion to take place.
Nuclear fusion is less efficient per interaction than plutonium fission, but hydrogen bombs more than make up for that relative inefficiency by carrying large quantities of hydrogen fuel. The greater number of reactions in a nuclear bomb drive its force from the kiloton range common for atomic bombs to the megaton range that has made nuclear weapons some of the most powerful devices ever built.