If you translate eudaimonia as "happiness," Aristotle's ethics collapses into something it was never meant to be. Happiness in modern English names a feeling — pleasant, fluctuating, available this afternoon. Eudaimonia names something structural: a life that is going well, taken as a whole, whether or not it feels pleasant at any given moment.
The distinction does real work. You can be in a good mood while your life is a wreck, and you can be having a difficult year inside a life that is unmistakably flourishing. Eudaimonia tracks the second thing. It is the reason Aristotle can say that a person's life cannot be fully assessed until it is complete, a remark that makes no sense at all if he is talking about feelings.
Aristotle's argument for it runs through function. Everything with a characteristic activity has a corresponding excellence: a good knife cuts well, a good eye sees clearly. What, then, is distinctively human? Not growth, which plants share, nor sensation, which animals share, but reason. So a human life goes well when it exercises reason excellently — in thought, and in the governance of character and action. Excellence of character is what he means by virtue, and virtue is therefore not a constraint on flourishing but the means to it.
This picture has a demanding consequence: flourishing is not entirely up to you. Aristotle acknowledges that eudaimonia can be damaged by catastrophe, poverty, or the loss of those you love — a concession the Stoics later rejected, insisting that virtue alone suffices for a good life regardless of circumstance. That disagreement between Aristotle and the Stoics is one of the sharpest in ancient ethics, and it turns entirely on how much luck a good life can absorb.
The concept has had a long afterlife. Contemporary virtue ethics rebuilt itself around it, and modern psychology borrowed the term to distinguish "eudaimonic" wellbeing — meaning, growth, purpose — from "hedonic" wellbeing, or pleasure and satisfaction. That distinction remains, at bottom, Aristotle's.