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The accepted atomic theory, that atoms are constructed from a small nucleus surrounded by electrons, has been very successful, despite being unable to explain how the nucleus itself behaves. A problem that confronted early atomic physicists was a method to keep the nucleus together. The nuclear forces have to be strong enough to overcome the Colomb repulsion between the protons.
Further theories were developed, and experiments performed to explain the nucleus. This culminated in the 1960s with the beginnings of the Standard Model.
The Standard Model describes the components of the nucleus, protons and neutrons, as being made up of quarks. The proton (neutron) is made up of three quarks, two up (down) quarks and one down (up) quark. 
It is not neutrons and protons that interact according to this model but rather, the quarks themselves that interact. The interaction takes the form of exchanges of virtual particles carrying the strong force. These virtual particles are called, appropriately, gluons.
The exchange of virtual particles is not unique to the strong force; force carrying particles are postulated to exist for all forces. Some are easily observed (the photon carries the electromagnetic force), while others (the gravitron carries the force of gravity) have yet to be observed.
The strong force is some times stronger than the electric force, yet it has an extremely short range - it barely manages to cross the nucleus of some larger atoms. This is because the gluon is a virtual particle having mass - it can only exist for a limited amount of time, and so has trouble exerting its influence beyond a distance of about . The infinite range of the electric force is permitted because photons are massless.