A nuclear phenomenon is simply something that happens in the nucleus of the atom.
Radioactivity occurs when the nucleus of the atom is unstable, so it releases some energy or matter within it to the outside, in the form of alpha, beta or gamma rays.
Instability of a nucleus is simply where it isn't stuck together right - there may be too many nucleons (protons and neutrons) - in which case the nucleus emits an alpha particle of two protons and two neutrons - or too many of one sort of nucleon, in which case a neutron can decay into a proton, or vice versa, releasing a beta particle. Or it may simply have too much energy, in which case a gamma ray is released.
Overall, though, it is called a nuclear phenomenon because it only involves the nucleus.