The strong nuclear force, one of the four fundamental universal forces, is responsible for holding atoms together. Without it, the nucleus, consisting of positive protons and neutral neutrons, would have nothing to stop the like charges of neighboring protons (positive/positive) from repelling one another.
The strong nuclear force consists of neighboring protons in the nucleus exchanging mesons, subatomic particles made up of quarks and anti-quarks that decay to form protons, neutrons and electrons. Charged mesons decay to form neutrons and electrons, and non-charged mesons decay to form protons.
The other three fundamental forces of the universe are the weak nuclear force, the gravitational force and the electromagnetic force.