What do protons and neutrons, the nuclear particles, do with respect to atomic stability?
They participate in the strong nuclear force which binds atomic NUCLEI together
The strong nuclear force is a strong (duh!), SHORT RANGE force, that binds the nucleons (protons and neutrons) together in atomic nuclei. This force is strong enough at short ranges to overcome proton-proton repulsion such that a net attractive force results. You might get a more sophisticated answer in the Physics forum, because this force is one about which chemists generally don't have much of a clue.
Why should protons repel each other in the nucleus?