The nucleus of an atom is made up of protons and neutrons. Both have about the same weight although protons have a positive charge and neutrons have no charge (they are neutral). Like charges such as protons produce a repulsive force against each other. This then begs the question, what holds the nucleus together then if these positively charged protons are all pushing each other away while being crammed into the nucleus together?
In truth, the atom is a truly lovely concept, full of answers to so many physical properties of everything we see around us. This goes all the way down to the tiny nucleus which although it contains almost all the mass, it takes up almost no volume of the atom because it is so tiny. When you pack lots of positively charged protons into the atomic nucleus, the nucleus does experience the force of repulsion from all the positively charged protons. What can also be found there is the strong nuclear force. The strong nuclear force happens to be much stronger than the electric force which in turn enables the nucleus to stay together. The neutrons in the nucleus do not produce an electric force like that of the electrons or the protons but they do produce the strong nuclear force and contribute to the binding energy of the nucleus.
One fun part of this is the name given to the mediator of this force. In a bit of humor if not for lack of a better name, the strong nuclear force is the glue that holds protons together so the name used for the particles is that of a gluon (pronounced glue-on). Just as the electric force is accurately described by the quantum mechanical process of exchanging photons, the strong nuclear force is described by exchanging gluons.
There is a limit to what this strong nuclear force can do though, if too many or too few neutrons are in the nucleus, the atom will be unstable and undergo radioactive decay. Generally you want to have almost as many protons as you have neutrons in the nucleus of an atom, just a fraction more neutrons than protons is generally the right mix for a stable nucleus. Another way to look at this would be to say that too many neutrons is like having too few protons or alternatively too few neutrons is the same as having too many protons.
If the nucleus had too many protons, then this excess positive charge can be compensated by absorbing an orbiting electron which is negatively charged into the nucleus leaving the overall atom in the form of an ion having one too few electrons attached to it. There are other more complicated ways an atom with too many protons can decay such as the emission of antimatter but having too few protons can result in completely different kinds of radioactive decay.
The harsh reality is that just as scientists are looking for evidence that gravity is chunked up into quanta of gravitons, they have never seen direct evidence of a gluon. Rather, just as it is clear that mass attracts other mass through what we call gravity, the force holding the nucleus together is what we call the strong nuclear force. Whether gravity does turn out to obey the expectations of quantum mechanics and to be quantized like other forces, the strong nuclear force does appear to follow the rules of quantum mechanics rather precisely.