Fission and fusion are two ways to release energy from atomic nuclei via nuclear reaction. The difference between them is in the process: One melds atoms with smaller nuclei together by fusing them while the other breaks them apart into fission products. In either case, the amount of energy involved is so large, millions of times more than from other energy sources, that these nuclear processes only happens in specific conditions.
As a verb, fuse is synonymous with "combine" or "blend." It follows that in a nuclear fusion process, two light nuclei fuse together to form a heavier nucleus. For example, two hydrogen atoms can fuse together to form one deuterium.
Tremendously high energy, usually in the form of extreme heat creating very high temperatures, and pressure is required to coax two strongly positive nuclei that would normally repel into a close enough space for fusion to occur, releasing nuclear energy in the process.
As a result, this process only happens inside stars like the sun which have a natural fusion reactor in their cores. Humanity can temporarily create the conditions for nuclear fusion, for instance with a hydrogen bomb, but sustaining such high temperatures necessary for a controlled, ongoing reaction to use as an energy source is not yet possible.
Once nuclear fusion begins, however, it can continue in a self-sustaining chain reaction. This is because the smaller atoms with masses up to that of iron on the periodic table give off more energy when fused than is required to fuse them together (an exothermic reaction). As such, nuclear fusion is the process by which most stars give off energy.
Fission, which can be defined as the act of splitting something into parts, is the opposite of fusion.
In nuclear fission, a heavy nucleus breaks apart into lighter nuclei. The breakage occurs when a neutron slams into a heavy nucleus, creating very radioactive and unstable byproducts, along with more neutrons, which continue to break down in a nuclear chain reaction.
The energy released from nuclear fission is millions of times more efficient than that released from burning an equivalent mass of coal. Unlike fusion reactions, fission reactions are relatively easy to initiate and control inside nuclear reactors, making them a widespread source of energy.