Hey
Your getting into strange teritory here. For the purposes of you and me, its probably easiest to consider the nucleus of the atom in isolation when your talking about radioactivty and nuclear stability. Don't worry about the electrons, theyre off outside doing their own thing.
As mentioned, positively charged protons will repel each other due to the electrostatic force. However pairs of protons and/or neutrons are attracted at close range by the strong nuclear force. As the nucleus gets bigger and the strength of the proton-proton repulsion increases, a proportionally greater number of neutrons are needed to compensate by spacing the protons apart and allowing more strong nuclear interactions.
If you have a nucleus with too many neutrons, they may undergo beta decay to form a proton and electron instead. If there are too few, you can get electron capture, where the nucleus traps and electron and this combines with a proton to form a neutron. Alpha decay is good to reduce the mass of excessively heavy nuclei.
Thats as much as i can explain. As to why some nuclei can have too many neutrons, i dont know the answer to that. Ask a nuclear physicist :P
http://www.algebralab.org/practice/practice.aspx?file=reading_thebandofstability.xml