Neutrons are neutrally charged particles that are found in the nucleus of atoms, along with the positively charged protons. As you can see in figure 1, the protons are pushed apart by the electromagnetic force, but pulled together by the strong force, which is stronger at short distances (these distances are about a fm or 10-15 m). The strong force comes from the interactions among the protons and neutrons. Neutrons are incredibly small, about 10-15 m, 10,000x smaller than an atom! Please see size of the universe for some online demonstrations to show this scale.
The number of neutrons in a nucleus determines what isotope the nucleus is. Some isotopes are stable, and some aren't, depending on the ratio of protons to neutrons. If a nucleus has too many neutrons, it undergoes one type of beta decay and if has too few neutrons, it undergoes the opposite type of beta decay, see the bottom of the page for a PhET simulation about this stability. These decays are due to another force called the weak force.
To learn more about neutrons please see hyperphysics.
The University of Colorado has graciously allowed us to use the following Phet simulation. This simulation builds atoms from protons, neutrons and electrons and tests knowledge of the periodic table. The simulation shows how the neutrons and protons must balance for the nucleus to be stable.