The strong nuclear force is one of four fundamental forces in nature. The strong force is 'felt' between nucleons (protons and neutrons) inside of the nucleus of an atom. The strong nuclear force is sometimes referred to as just the strong force or the strong interaction.
This force is strong enough that it overcomes the repulsive force between the two positively charged protons, allowing protons and neutrons to stick together in an unimaginably small space. The strong force dies off with distance much faster than gravity or the electromagnetic force, so fast that it's almost impossible to detect the strong force outside of a nucleus. (The strong force and the weak force are not inverse square laws.)
The nucleus (and the distance over which the strong force acts) is incredibly small (please see size of the universe for some online demonstrations to show this scale). Despite these small sizes, they still produce a great deal of energy. As is discussed on the work page, the stronger the force (or the greater the distance), the more energy is transferred for an interaction. The strong force stores an incredibly large amount of energy in nuclei compared to the electromagnetic force, which is what governs chemical reactions. This is why nuclear fuel has ~1 million times the density of any chemical-based fuel (coal, natural gas, oil); see energy density for graphics displaying this difference. The big challenge is that very careful engineering is required to access the energy stored from the strong force.
A full treatment of the strong force requires many years of intensive study. To learn more about the strong force please see the hyperphysics article on the strong force. Additionally, there is a comprehensive (although longer) article on what holds nuclei together by Prof. Matt Strassler.
And here is part 1b.
To learn about the other forces, please see the following pages: