What is the difference between nuclear decay and nuclear fission?

• Nuclear decay is any spontaneous process where unstable nuclei release extra energy to arrive at a more stable state. Typical decay processes are Alpha, Beta, and Gamma. Some large unstable nuclei (e.g. Plutonium-240) can sometimes decay by spontaneous fission.
• Nuclear fission is a process where certain large nuclei (e.g. Uranium-235 & Plutonium-239) absorb a neutron and then split into two smaller nuclei and a few free neutrons. Some large unstable nuclei (e.g. Plutonium-240) don't need to be hit by a neutron to fission.
• Nuclear fusion is a process where small nuclei under unusual conditions of very high temperature and very high pressure combine to form larger nuclei.

All three processes above are exothermic.

In stars nuclear fusion stops at nickel and iron (further fusion past this would be endothermic). If all we had was the above processes then that would be where the Periodic Table ended (therefore there could not be nuclear fission as such heavy nuclei could not exist). However stars die, and some die so spectacularly we call them supernovas.

When a supernova occurs, an intense shock wave blows all the outer layers of the star away at very high velocity. At these velocities nuclei collide so hard that normally impossible endothermic nuclear fusion reactions occur. The rest of the periodic table is filled here, including many transuranics not found naturally on earth (e.g. Americium, Californium, Berkelium).