Why are some isotopes stable and others are not stable?

The nucleus contains protons and neutrons, positive and neutral particles. The positive protons are trying to repel each other and the neutrons provide a 'buffering' that holds the nucleus together (called binding energy), some theories say this involves the continuous exchange of quarks, that allows the protons to stay close to each other in the nucleus. Larger nuclei have a different problem, the nucleus is large enough that different regions, like the extreme top and the extreme bottom repel and can cause the atom to expel particles. This is radioactive decay. One scientist (George Gamow?) came up with a theory of constantly moving nuclei that, instead of being a static spherical nucleus, would act like a liquid drop that is undulating and the constant movement and change of shape could cause deformation to a point that the repulsion of opposite sides of the nucleus would cause the decay.

Short answer -- all those positive protons in a large nucleus just don't 'like' being that close to each other.