All forms of egoism require explication of “self-interest” (or “welfare” or “well-being”). There are two main theories. Preference or desire accounts identify self-interest with the satisfaction of one's desires. Often, and most plausibly, these desires are restricted to self-regarding desires. What makes a desire self-regarding is controversial, but there are clear cases and counter-cases: a desire for my own pleasure is self-regarding; a desire for the welfare of others is not. Objective accounts identify self-interest with the possession of states (such as virtue or knowledge) that are valued independently of whether they are desired. Hedonism, which identifies self-interest with pleasure, is either a preference or an objective account, according to whether what counts as pleasure is determined by one's desires.
Psychological egoism claims that each person has but one ultimate aim: her own welfare. This allows for action that fails to maximize perceived self-interest, but rules out the sort of behavior psychological egoists like to target -- such as altruistic behavior or motivation by thoughts of duty alone. It allows for weakness of will, since in weakness of will cases I am still aiming at my own welfare; I am weak in that I do not act as I aim. And it allows for aiming at things other than one's welfare, such as helping others, where these things are a means to one's welfare.
Psychological egoism is supported by our frequent observation of self-interested behavior. Apparently altruistic action is often revealed to be self-interested. And we typically motivate people by appealing to their self-interest (through, for example, punishments and rewards).
A common objection to psychological egoism, made famously by Joseph Butler, is that I must desire things other than my own welfare in order to get welfare. Say I derive welfare from playing hockey. Unless I desired, for its own sake, to play hockey, I would not derive welfare from...