Eudaimonia is a Greek word that consists of eu (felicity) and daimon, which was a small spirit which supposedly dwelled on your shoulder throughout life overseeing your actions. Hence, eudaimonia essentially means the felicity of your spirit, or the state of living well and of having lived well throughout your entire life.
Although commonly translated as "happiness", happiness is a state that can come and go. You might be happy one minute and miserable the next. Eudaimonia means something more than this. It is not a state that can come and go, but means to have lived well throughout the course of your life, just as we might identify one man as a particularly "lucky man" whilst others just enjoyed brief periods of luck. Hence, eudaimonia can only really be recognized after an individual's death, when it becomes possible to sum up and pass judgement on their life as a whole, just as it is not possible to truly say someone is or was a good man until he has died and his capacity for wrongdoing has hence come to an end.
This has led some people to suggest that becoming eudaimon is something which can only be brought at the price of your life, through summing up your existence in your greatest deed. For instance, using the story of Achilles as an example, Hannah Arendt points out that his premature death was a precondition of his greatness; only by summing up his existence in one act could he prevent this act being diluted by the decades of mediocrity which were bound to follow and remain truly eudaimon.