An atom consists of a nucleus of protons and neutrons, which from the nucleus (thus also called nucleons). The nucleus is surrounded by electrons.
Two protons in the same nucleus do repel each other but the repulsion is overcome by the strong nuclear force.
Isotopes that contain unstable nuclei are called radioisotopes. Unstable isotopes can emit various types of radiations.
The strong nuclear force is about 137 times stronger than electromagnetic forces of the charges. The strong nuclear force between two nucleons is strongest at around 1e-15 metres, and disappears at about 2.5 times this distance.