The forces that hold atoms together are the electrical force and the strong force, which is stronger than the electrical force. The electrical force does the majority of the work of holding atoms together, but the strong force helps hold in the electrical force and can somewhat override it.
The electrical charge that is exhibited from protons and electrons attracting to each other causes the force of the atoms to become stronger. The charge sets up the framework for the electrical force that is used to hold everything together, but it is not strong enough to keep the atoms close enough together. While the force helps electrons and protons attract to each other, it does nothing for the neutrons, which are neutral. It also works against itself because two protons and two electrons repel each other, even though they are pulling closer together through the electrical charge.
The strong force is able to remedy the problem of the electrical charge not being strong enough to hold everything together. It is stronger than the electrical force and helps to hold all of the protons and neutrons in, which are immune to the electrical force. It is a very strong force, but it only works in small ranges, making it ideal for holding atoms together. It is also able to override the repulsion that is sometimes caused by too many positive charges in an electrical force.