covalent bond
Suppose two atoms both want extra electrons and neither is willing to simply give any away — for instance two hydrogen atoms, or two chlorines. Instead of transferring electrons, they compromise by sharing a pair: each contributes one electron, and the shared pair belongs to both nuclei at once. That shared pair sitting between the two nuclei is a covalent bond, the glue of almost every molecule, from H2 to DNA.
What actually holds the atoms together is that the shared electrons spend time in the region between the two positive nuclei, where they attract both nuclei and screen their mutual repulsion. The bond settles at the distance where attraction and repulsion balance — the bond length. A single bond is one shared pair; atoms can also share two pairs (a double bond) or three pairs (a triple bond, as in N2), giving shorter, stronger bonds. Each shared pair can be pictured as a region of overlapping atomic orbitals.
Covalent bonding produces discrete molecules (or extended covalent networks like diamond) and is the basis for drawing Lewis structures, predicting shapes with VSEPR, and the more detailed valence-bond and molecular-orbital pictures. When the two atoms differ in electronegativity the sharing is unequal — a polar covalent bond — which gives the molecule a dipole. So covalent and ionic bonding are two ends of one continuum: equal sharing at one end, complete transfer at the other.
In H2 each hydrogen brings one electron; the shared pair sits between the two protons, so both nuclei feel as if they have a filled 1s shell. In N2 the two nitrogens share three pairs, a triple bond, which is why N2 is so short, strong and famously unreactive.
Sharing a pair (H2) versus sharing three pairs (the N2 triple bond).
A covalent bond is not always two atoms each donating one electron — in a dative (coordinate) bond one atom supplies both electrons of the shared pair, as in the N-to-B bond of H3N-BF3; once formed it is an ordinary covalent bond.