What is the primary force that holds the protons and neutrons together within the nucleus, despite the electrostatic repulsion between protons?

The primary force is the strong nuclear force (or strong interaction).

Protons, being positively charged, naturally repel each other. The strong nuclear force is an extremely powerful attractive force that acts over very short distances, overcoming this electrostatic repulsion to bind the nucleons (protons and neutrons) together in the nucleus.