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Protons are subatomic particles with a positive charge that make up the composition of the atomic nucleus.
A proton is a positively charged subatomic particle with a mass of approximately 1 AMU, slightly less than a neutron. The atomic number of an atom is based on the number of protons in its nucleus.
The proton that is important to chemists comes from the ionization of the hydrogen atom. The resulting ion completely lacks electrons and consists only of this subatomic particle; because of its tiny size it doesn't exist as a stand-alone ion and sticks to molecules and anions, producing cations and neutral molecules respectively.
Protons are donated to other molecules by compounds known as acids (such as sulfuric acid, hydrochloric acid, etc), and may be received by compounds known as bases (sodium hydroxide, pyridine, etc). This process of transferring a proton is known as protonation. During protonation, a base will conduct a nucleophilic attack on an acid, pushing electrons away from the proton, and producing the conjugate base of the acid.