Some nuclei are fissile (but only a few of the most massive), some are neutron-rich and tend to emit #beta# particles, some have an excess of protons (positron emitters.) Have a look a time the diagram below and you can see the pattern:
Taken from: https://en.m.wikipedia.org/wiki/Stable_nuclide
A relatively large number emit #alpha# particles and this brings me to the point ... all nuclides are attempting to become stable through these transitions, because this means lowering their internal energy. Only the (relatively few) black dots are stable on this diagram!
Emitting #alpha# particles is harder to explain using this model, but an alternative way of looking at the nucleus may help explain it (the liquid drop model of the nucleus.) Because an #alpha# particle is so stable it is relatively likely to “pinch” off a nucleus.
The final point I’d like to make is that very often a nucleus, having emitted a decay product or undergone fission, is left in an excited state, it will then emit a #gamma# ray as a means of shedding the excess energy.
Sorry it’s been such a long answer!