An example of ethical egoism would be a person who owes money to a friend and decides to pay the friend back not because that person owes money, but because it is in his best interest to pay his friend back so that he does not lose his friend. Another example of ethical egoism would be a person who invites a friend to a movie that she wants to see because she does not want to go alone and is thinking of her own self-interests first.
In ethical egoism, a person focuses on her own self-interests before all others. Ethical egoism proponents believe that focusing on one's own self-interests benefits all of society because society runs more efficiently. Thomas Hobbes was a famous proponent of ethical egoism.