Fusion (specifically, nuclear fusion) is a type of nuclear reaction, specifically, the formation of an atomic nucleus from smaller nuclei. Some fusion reactions release energy in the form of electromagnetic radiation and/or particles such as neutrinos, while others require energy to take place. Fusion (the former kind, releasing energy) is the primary source of energy in the long life of stars and is a factor in explosions such as supernovae. In both cases, it constitutes nucleosynthesis as well.
The opposite nuclear reaction, division of a nucleus into multiple nuclei, is nuclear fission (or just fission). In both cases, certain conditions make the event more likely: for fusion, in addition to the particular starting particles for the particular reaction, the particles must have enough kinetic energy in the form of heat. The requirements for fission are similar, but the kinetic energy is needed in an extra incoming particle (e.g., neutron) to trigger the fission event.
In both cases, a new nucleus (or nuclei) results, of a different element (or elements), i.e., with a different atomic number (number of protons), and a different mass number (number of nucleons, i.e., protons and neutrons). The actual atomic mass is approximately the mass number, but is actually a touch larger or smaller because of the binding energy, i.e., the energy that holds this particular nucleus configuration together, which varies with the combination of atomic number and mass number. It is the difference in required binding energy for the initial nuclei and the final nuclei, when it results in an excess of energy, that produces fusion or fission power. A new configuration that requires more binding energy than the initial nuclei draws kinetic energy from the collision (and requires that much kinetic energy and thus a sufficient temperature).
In astrophysics, fusion is often informally referred to as burning, i.e., a star "burning hydrogen" or hydrogen burning to mean hydrogen nuclei fusing to produce something else. The word burning is being used metaphorically: hydrogen fusion is something like the process of chemically combining with oxygen to produce a different compound, but this burning is at the nuclear level: a nuclear process that produces a different nucleus and element.