How does the behavior of an electron change when it forms a covalent bond?

Answer: Covalent bonds are a convenient way of describing the way that a few negatively charged electrons can fix their average location so as to hold two positively charged nuclei together. Chemists know about covalent bonds, electrons do not. All electrons are constantly in very fast motion, they are all indistinguishable from one another, and we have to think in terms not of the position of each electron specifically, but of the average distribution of electrons in a molecule. "Bonds" and "localized electrons" are a convenient fictional calculus and accounting system that allows chemists to work out the properties of molecules and the mechanisms of chemical reactions.

Think of a baseball team made up of nine identical siblings, who constantly and rapidly change fielding positions. You can still describe the team as having a short stop or a second base man, even though you cannot identify an individual person with any role.