What is the force that holds quarks together inside protons and neutrons?

The force that holds quarks together inside protons and neutrons is the strong nuclear force, more precisely called the strong interaction.

Quarks carry a type of charge called colour charge, and they interact by exchanging gluons, the force-carrying particles of quantum chromodynamics. This interaction confines quarks: they are not observed as freely isolated particles under ordinary conditions. Instead, they remain bound into composite particles such as protons and neutrons.

The strong interaction is especially powerful at subnuclear distances, roughly below a femtometre. An important subtlety is that the force binding quarks inside a proton is not exactly the same as the residual strong force between protons and neutrons in an atomic nucleus. The latter is a leftover effect of the deeper quark-gluon interaction.

The strong interaction also explains most of a proton’s mass-energy. The quarks themselves contribute only a small fraction; much of the mass comes from the energy of their motion and binding fields.