Your question boils down to the interactions among subatomic particles, which I'll do my best to explain in a concise yet satisfactory manner.

At a fundamental level, there are four known forces that describe how all of the stuff interacts in the Universe:
1) Strong aka Nuclear,
2) Electromagnetism,
3) Weak
4) Gravity
in order of decreasing relative strength. There is a certain type of "exchange particle" associated with each of these forces. For example, the Electromagnetic (EM) force between two charged particles is described by the exchange of photons. As you may know, particles with similar charges (++ or --) repel each other, while oppositely-charged particles (+-) attract.

Now take a nucleus composed of protons (+) and neutrons: based on the description above for the EM force, you would expect the protons to repel each other at close distances, especially since the strength of the EM force is proportional to 1/r² (where "r" is the distance between the protons). But the nucleus stays together...how is that possible? Intuitively, there must be some additional force present that is strong enough to overcome the EM repulsive forces between protons. This brings us to the Nuclear force, which you can think of as responsible for holding the nucleus together.

Here's where the story gets more complicated-
protons and neutrons are not fundamental particles, but in fact made up of smaller (elementary) particles called quarks. According to the theory developed by Professors Moo-Young Han and Yoichiro Nambu in 1965, quarks possess a property called color, or color charge. You can think of this color (red, green, or blue) as analogous to charge (+,-) in the EM force, where like colors repel and opposites attract. However, the use of the term color here does not imply anything about the visual appearance of the quark, but rather describes how the three different quantum states of the quark can mix (i.e., red + green + blue = "colorless"). Much like the exchange of photons between two charged particles with the EM force, gluons are the "exchange particles" for the Nuclear force between quarks.

There are not one but two nuclear forces, the so-called weak force, and the strong/color force. Both of these are fundamental forces of nature, like electromagnetism and gravity. So far physicists don't understand why it is that any of the four forces exist, although there are some similarities between the four forces and at least some of the forces merge together at high energy levels (the weak force in particular merges with the electromagnetic force).

My understanding of the weak force is somewhat limited, but from what I do understand it drives radioactive decay, at least radioactive decay where one of the products of the decay is an electron or positron (and neutrinos). I also believe that the weak force is responsible for matter-antimatter annihilation, since that also produces both photons (namely, gamma rays) and neutrinos.

The strong force is much more important and easier to understand; the theory that describes it is Richard Feynman's theory of quantum chromodynamics (for which Feynman got the Nobel Prize). All four forces are conveyed by interactions involving a kind of charge, be it mass (gravity), electric charge (electromagnetism), or what is referred to as "color" for the strong force (which is why it is also referred to as the color force). In the strong force, as with the electromagnetic force, like charges repel and unlike charges attract, but unlike electromagnetism, there are three different kinds of charge in the strong force, normally referred to as blue, green, and red, which is why it's called the color force (note that blue, green, and red, in this case, have nothing to do with blue, green, or red light - light is an electromagnetic phenomenon, not a nuclear phenomenon). Also like the electromagnetic force, in which no two electrons in the universe can have the same spin (another quantum property), energy, and position, the same is true of quarks, which are the particles that the strong force acts on. This is why you can't normally pack protons too closely together, since protons are made of quarks, but it's also what keeps the quarks in the protons from flying apart, because they have opposite colors (specifically, one blue, one green, and one red, in every proton). Neutrons, which are also composed of quarks, are governed by the same principle. Atomic nuclei composed of multiple protons and neutrons are held together by much weaker interactions between quarks in neighboring protons or neutrons, in a fashion similar to Van der Wall's forces in hydrophobic chemicals (which are electromagnetic in nature).

At higher energy levels, the electroweak force appears which is the unification of the electromagnetic and weak forces, and is carried by a new set of force-mediating particles that only exist at those higher energy levels. Theories predict that the color force will merge with the electroweak force at even higher energy levels, but so far this has never been seen in any experiment, and several predictions of popular theories of how this merger would occur have not been observed, despite efforts to find them (namely, there are no magnetic monopoles that have ever been found, and we have never seen a proton decay either). Neutrinos do have mass, though, which was predicted by this "grand unification" theory, and people are continuing to look for the other missing phenomena.