A self-centered person is excessively concerned with himself and his own needs. He's selfish.
You probably know some people who always talk about themselves, make every issue about themselves, and are generally all about "Me, me, me!" People like that are self-centered: as the word suggests, they are overly centered on themselves. Self-centered people tend to ignore the needs of others and only do what's best for them. You can also call them egocentric, egoistic, and egoistical. In the seventeenth century, however, self-centered meant "fixed or stationary."