There are forces in the nucleus that oppose each other, the Strong force holding Protons and Neutrons to each other and the electrostatic force of protons repelling other protons. Under certain arrangements of protons and neutrons the electrostatic force can cause instability in the nucleus causing it to decay. It will continue to decay until it reaches a stable combination.
The observed forms of decay are alpha decay, beta decay, Electron capture, neutron emission, positron emission, proton emission, and spontaneous fission. The latter five forms of decay occur very quickly within products of nuclear reactions, and hence are not often seen on earth outside a nuclear reactor. By contrast alpha and beta decay are seen in the decay chains of radioactive materials.
Neutron emission is also important as the most important reason for the difficulty of manufacturing a nuclear bomb from lower grades of plutonium.
Radioactive decay is observed astronomically in supernova, and the light curve of supernova is generated via the decay of radioactive nickel into iron.
Many radionuclides have several different modes of decay, each with its own probability. Bismuth-212, for example, has three.
All radioactive decay is also associated with emission of gamma radiation in varying degrees.
Nearly all decay products are themselves radioactive, giving rise to decay chains which eventually end in a stable nuclide.