Why do atoms bond, and what is so special about eight electrons?

Atoms bond because energy is released when they do, and the clue to how is in the electrons. The noble gases barely react, and Walther Kossel and Gilbert Lewis saw that this was the clue. They proposed that during the formation of a chemical bond, atoms gain, lose or share electrons in such a way that they acquire the nearest noble gas configuration.

Electrons sit in shells, and each shell holds a fixed number: two in the first, eight in the second, 18 in the third, following 2n squared. The octet rule says that main-group atoms tend to bond so that each has eight electrons in its valence shell, like a noble gas atom. The third shell can hold 18, but its extra room sits at considerably higher energy, so eight already counts as a closed shell for chemistry.

Table salt shows the transfer way. Sodium transfers one electron to chlorine, and the result is a lattice of Na+ and Cl- ions. Carbon dioxide shows the sharing way: each oxygen shares four electrons with the carbon, and shared electrons count in the octets of both atoms.

It is a rule of thumb, not a law. Hydrogen and lithium follow a duet rule and tend to have the same two-electron configuration as helium. Lewis, the man who discovered the covalent bond and the electron pair, was nominated 41 times for the Nobel Prize in Chemistry and never won.