How do valence electrons form bonds?

Covalent bonding occurs when an electron from each of the bonding atoms is donated to an electron pair. This forms a type of covalent bond called a single covalent bond. Each bonding atom can also donate more than 1 electron to multiple electron pairs in a double/triple etc. covalent bonds. Covalent bonds can also be dative; both electrons of the electron pair are donated by only one of the bonding atoms. What makes the bond a 'bond' is the relatively strong electrostatic force of attraction between the positive nuclei of the bonding atoms and the negative electron pair(s).

More specifically, covalent bonding can form sigma bonds or pi bonds. Sigma bonds are a result of the direct overlap of atomic orbitals between two bonding atoms. This forms a single covalent bond. Pi bonds are a result of the sideways overlap of the p-orbitals between the two bonding atoms. This forms a double covalent bond. Electron density in a sigma bond is concentrated along the plane of the 2 bonding atoms and in a pi bond, it is concentrated above and below the plane.

Ionic bonding is where electrons are donated and accepted to form cations and anions. The oppositely charged ions then form an ionic bond between them due to the relatively strong electrostatic forces of attraction between the two.