A nuclear fission bomb uses two main features: first, that uranium is a radioactive material which spontaneously breaks down into two smaller elements, at the same time releasing a small amount of energy and two or more neutrons.
Second is the concept of a chain reaction. In a chain reaction a neutron strikes the nucleus of a uranium atom causing it to split, releasing two or more additional neutrons. Each of these neutrons can then hit other uranium atoms causing them to split and release even more neutrons.
So start with one neutron hitting one nucleus and then releasing two neutrons. These two neutrons hit two more nuclei releasing a total of 4 neutrons, which hit 4 more nuclei, releasing 8 neutrons, and very quickly you have a very large number of uranium atoms being split.
In a nuclear power plant this chain reaction process is controlled so only a certain number of neutrons are flying around at a time. In a fission bomb a number of pieces of uranium are brought together all at once and an uncontrolled chain reaction occurs, releasing a tremendous amount of energy all at once.
A fusion bomb works on the principle that if you force the nuclei of two small atoms close enough together, you can overcome the natural repulsion between their positive charges and form a new, larger element. Thus, if you combine two hydrogen atoms, you can make a helium atom. In the process of fusing the two atoms a small amount of energy is also released.
Fusion is a much more difficult process to achieve because it requries very high temperatures and pressures to force atoms to join together. In fact, a fission bomb may be required to provide enough energy for the fusion, or hydrogen bomb, to ignite.
In summary - a fission bomb relies on the natural, spontaneous breakdown of large radioactive elements, especially uranium, into smaller elements through a chain reaction. You can't stop fission from occuring but you can control how fast it happens.
A fusion bomb requries large amounts of energy and very high pressures and is not a natural process on earth. You have to force it to ignite.