What is the name of the force that holds atomic nuclei together?

The force that holds atomic nuclei together is the strong nuclear force.

More precisely, the attraction between protons and neutrons inside a nucleus is called the nuclear force or residual strong force. It is a leftover effect of the more fundamental strong interaction, which binds quarks together inside protons and neutrons. This short-range attraction keeps nucleons packed into a nucleus.

The nuclear force must overcome the electromagnetic repulsion between positively charged protons. It is strongly attractive at distances of roughly 0.8 femtometres, but it fades rapidly beyond about 2.5 femtometres. At extremely short distances, it becomes repulsive, helping prevent nucleons from collapsing into one another.

A common mix-up is confusing the nuclear force with the strong interaction itself. The strong interaction directly confines quarks; its residual effect binds whole protons and neutrons into nuclei. The weak nuclear force instead causes processes such as beta decay.