The strong nuclear force or strong interaction is a fundamental force of nature which affects only quarks, antiquarks, and gluons. This force is responsible for binding quarks together to form hadrons (including protons and neutrons), and the residual effects also bind these neutrons and protons together in the nucleus of the atom. See particle physics for an overview of the theory.
According to quantum chromodynamics, every quark carries color charge which comes in three types: "red", "green" and "blue". These are just names and not related to ordinary colors in any way. Antiquarks are either "anti-red", "anti-green" or "anti-blue". Like colors repel, unlike colors attract. The attraction between a color and its anti-color is especially strong. Particles can only exist if their total color is neutral, meaning that they can either be composed of a red, green and blue quark (such a particle is called a baryon; protons and neutrons are examples), or of a quark and an anti-quark having the corresponding anti-color (such a particle is called a meson).
The strong interaction acts between two quarks by exchanging particles called gluons. There are eight types of gluons, each carrying a color charge and an anti-color charge.
As pairs of quarks interact, they constantly change their color, but in such a way that the total color charge is conserved. If say a red quark is attracted to a green quark inside a baryon, a gluon carrying anti-green and red color is emitted from the red quark and absorbed by the green quark; as a result the first quark switches to green and the second to red (total color charge remains green + red). If a blue quark and a anti-blue antiquark interact inside a meson, a gluon carrying for example anti-red and blue could be emitted by the blue quark and absorbed by the anti-blue one; as a result the blue quark turns red and the anti-blue antiquark turns anti-red (total color charge remains 0). Two green quarks repel each other by exchanging a gluon carrying green and anti-green color; the quarks remain green.
Unlike the other fundamental forces, the strong interaction also acts on the strong exchange particles themselves, since gluons carry color charge. This leads to a very limited range of the strong interaction (not much farther than the hadron's radius) even though the gluon does not have mass. It also has the strange effect that the force gets stronger as the distance between the quarks increases. This effect prevents free quarks from being observed. As the distance between two quarks increases, the amount of energy in the force between them increases. If the force becomes strong enough, there is enough energy to create new quarks. This is the reason that one only sees quarks in pairs or triplets and never individually. The textbook allegory is that of a rubber band. When the rubber band is stretched far enough, the band breaks and you have two new rubber bands. Similar with quarks: separate the quark pair far enough, and two new quarks will pop up.
The phenomenon of not being able to separate quarks, is called confinement. It is conjectured as the quarks are moved really close, the quarks no longer interact via the strong interaction, and become `free' - this is called asymptotic freedom. The allegory of the rubber band holds here too. Move the ends of the band close together, and they do not `feel' each other.