The strong interaction or strong force is one of the four fundamental interactions or forces of nature. It is called the strong force because the other three forces – electromagnetism, weak interaction and gravitation – are considerably weaker. Electromagnetism is about 100 times weaker, but weak interaction and gravitation don't even come close with former being a million times and latter about 10^39 weaker.
The strong interaction does two things:
First, it binds together fundamental particles named quarks. Quarks have charges called colour-charges. There's six of them, red, blue and green, each both positive and negative. These charges have nothing to do with colours, it's only a way to differentiate them. Think of them like the strong interaction's version of electromagnetism's two charges. By binding quarks together, the strong force creates hadrons. Hadrons is the name for all particles that are made of two, three or hypothetically more quarks, but to keep it simple, just have in mind that protons or neutrons are both made of three quarks each and thus best-known hadrons. By keeping quarks together to form protons and neutrons, two of the largest components of the mass of ordinary matter, the strong interaction makes sure ordinary matter is stable.
Second, the strong interaction binds protons and neutrons together, making an atomic nucleus. Since protons have positive electric charge and neutrons no electric charge, without the strong force, atoms of elements other than hydrogen wouldn't be able to exist. We know that two positive electric charges repel each other, so atomic nucleus with two or more protons would not be able to exist. Being 100 times stronger than electromagnetism, the strong interaction easily overpowers it to keep nuclei of atoms stable.
There's a distinction between these two aspects of the strong force.
In first case, the strong force doesn't get weaker with distance, unlike other three fundamental forces. This is why physicists haven't been able to observe free quarks. Even if they tried to separate and isolate an individual quark, the amount of energy needed to pull two quarks apart would create new quarks that would pair with first two. This doesn't mean that quarks in one hadron (i.e. proton or neutron) attract quarks in another hadron. They now act as a single particle. When bound into a hadron, quarks' colour-charge cancels out and strong interaction between hadrons equals nearly zero.
In the second case, the strong force is not as strong as in the first case and it's actually called the residual strong force. It's strength is only the "leftover" from the force that binds quarks together, just like its described above. The residual strong force gets weaker with distance. For this reason, elements with large nuclei are unstable.
Carrier particle of the strong force is gluon. Like quarks, gluons can't exist alone and have never been observed to be free. In particle accelerator, when i.e. a proton is hit with another particle, it breaks apart, but the energy is not released in form of gluons, but new massive particles are produced. Three quarks in proton only make up 1% of the mass of the whole proton. The rest comes from the energy within the proton, held together by the strong force represented by gluons.