A person who has become a master of his environment. A gangster gets what he wants when he wants it. It is someone who has figured out the system and and uses it to achieve his goals. Typically a gangster turns to illegal means of making money because they are more profitable. A gangster has a very high opinion of himself because he has built his life and maintains his lifestyle through his own means.
Being a gangster is not about being a violent criminal. In fact, gangsters prefer not to be violent because it often gets in the way of profits. Gangsters are not afraid to become violent when necessary though.
Respect is the second most important currency to a gangster, behind money. A true gangster gives respect, and demands it in return.
In short, a gangster is a pure capitalist, using his own resources to get what he wants.
If you can think of a famous gangster you are most likely either wrong, or thinking of an unsuccessful gangster.