It is the electron.
The electrons are the most "moveable" for a given element, so they are involved in bonding. Especially valence electrons, or the electrons on the outside of an atom.
However, the proton also plays a role in bonding because of its positive charge, which attracts electrons.
A chemical bond results in the sharing of electron density between two atoms. The more complete this sharing of electron density is the stronger the bond. A single sigma bond is stronger than a single pi bond but not stronger than the resulting double or triple bond. This is because in a sigma bond the electron orbitals directly overlap making the electron density sharing more complete.
By sharing electron density atoms can achieve greater stability. The chemical bonds almost always result in a an electron configuration that is more stable than the electron configuration before the chemical bonding occurred.