Protons are the positively charged particles that are inside the nucleus of an atom.The protons are pushed apart by the electromagnetic force but pulled together by the strong force, which is stronger over short distances (these distances are about a fm or 10-15 m). Protons 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. Despite their incredibly small size, protons push against each other with tremendous forces, about 100 N, comparable to the weight of a small dog!
The charge on a proton is exactly equal and opposite to the charge on an electron. So the number of electrons in a neutral atom is always equal to the number of protons (see the bottom of the page for a PhET simulation about that). Protons are made made of smaller particles called quarks (or see hyperphysics), which also make up neutrons.
The number of protons in a nucleus is called the atomic number, and this number determines what element an atom is. In other words, changing the proton number changes the element. This number of protons (atomic number) changes when a nucleus undergoes beta decay in any of its various forms, or alpha decay. The difficulty with changing the number of protons in a nucleus on purpose is why alchemy (the medieval practice of trying to turn lead into gold) failed for so long!
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.