What are the forces that exist inside an atom?

The strong nuclear force, and the electromagnetic force.

Explanation:

There are two forces inside the nucleus of an atom who keeps it together. The strong nuclear force, and the electromagnetic force. The strong nuclear force is the one who keeps the protons and neutrons together despite the electric repulsion force of the particles, to understand this better, you have to remember that the atom nucleus has to elemental particles: protons and neutrons. Protons have a positive electric charge, and neutrons don't have any charge. So as you must know, electric charges who are similar tend to repulse when they are near each other, this is the electric repulsion force. But as the strong nuclear force is greater than electric repulsion force, due to the short distance of the particles inside the nucleus, it can overcome this repulsion to keep the protons "glue" inside the nucleus of the atom.

There are four forces: Electromagnetic, Strong, Weak, and Gravity.