What is Arete?
In ancient Greece, arete meant "goodness" or "excellence". It's related to "araomai", to pray, and to "aristos," meaning "the best", which is where "aristocracy" comes from: aristocracy is the rule of the best people. So having arete was necessary for being an aristocrat.
Ancient Greek people thought of arete as meaning something like the US Army's old slogan, "Be all that you can be." It meant that you were being the best person you could be. So arete would mean different things for different people. In the Odyssey, Penelope has arete, because she is the best wife that a woman can be. But Achilles in the Iliad also has arete, because he is the best warrior that a man can be, and Odysseus has arete because he is so clever, and thinks up effective plots, and athletes have arete when they win the foot-race.
Not only people could have arete - a well-built house, a beautiful piece of pottery, and a strong horse all had arete too. In Plato's Allegory of the Cave, the ideal form of a thing is its arete, the goal that everything is trying to get to. As Plato says, arete is something you are always trying to achieve, but it's always out of reach: it's a goal.