Nemesis started out as an accountant's word. Before it named a goddess or an enemy, it described the act of handing out shares.
The Greek verb nemein meant to deal out or allot, and in particular to give a person what was due to them. Nemesis was the noun: the distribution of fortune, good and bad, according to what each person deserved. From that came a second sense, "righteous anger," the feeling you have when someone gets more than they should.
The Greeks made that feeling into a goddess. Nemesis punished arrogance toward the gods, which the Greeks called hubris, and she balanced out people whose luck had run too long. She was often shown with a sword, scales or a measuring rod, tools for judging and correcting. Her best-known sanctuary was at Rhamnous in Attica. Her statue there is traditionally credited to Pheidias and was made after the Battle of Marathon, from marble the Persians had reportedly brought to build a victory monument.
Her family tree varies by source. In Hesiod's Theogony, the goddess Night bore her alone. Other writers give different parents. She also turns up in the story of Narcissus, whom she punishes in some tellings for rejecting those who loved him.
English took the name in the 1570s, still as a goddess. By the 1590s a lowercase nemesis meant retributive justice in general, as in the idea that wrongdoing brings its own punishment. The modern sense is much later. By 1930 it described an opponent or obstacle that seems certain to defeat you, a rival with a kind of fated edge.
That drift keeps the original logic. A nemesis is someone who seems to hold what is coming to you. The idea of cosmic balance has shrunk to a personal rivalry, but the sense of an account to be settled is still there.
The root nem- is also behind a surprising number of English words, including economy, nomad and astronomy.