What holds nonpolar covalent bonds together?
The attraction between the bonding electrons and the nuclei of the two atoms involved.
All bonds are effectively electrostatic attraction forces between oppositely charged species (electrons, nuclei or ions, typically). Covalent bonds are directional. This means the electrons are, in simple terms, effectively confined to the orbital i.e. the 'space between the two atoms which are interacting. In this position, they are attracted to and attract the positively charged nuclei and this force is the 'bond'. whether the bond is polar or non-polar does not change the fundamental argument.