Think of it like this: two atoms both "want" a full outer shell of electrons (like a noble gas). Instead of one atom stealing electrons from the other (that's ionic bonding), covalent bonding is more like a compromise — both atoms bring one electron each to a shared "electron pot" in the middle, and both atoms count those shared electrons as their own. Everyone's happy, nobody actually loses anything.

Formally: a covalent bond is the electrostatic attraction between the nuclei of two atoms and the shared pair of electrons in their outer shells. It forms because two atomic orbitals overlap to create one molecular orbital — and the electrons in that shared space are more stable being attracted to two positive nuclei instead of just one.

One subtlety that trips people up: don't picture the two shared electrons sitting still, frozen exactly halfway between the nuclei like two dots glued in place. In reality they're in constant motion and are better described as a charge cloud — a smeared-out region of negative charge that both nuclei are attracted into.