Thinking too well of yourself — believing you are more important, more talented or cleverer than you actually are, and showing it. The picture behind the word is a head that has swollen up with pride until it is too big. It is informal, mildly insulting, and slightly old-fashioned; it is mostly American, and it often turns up when success has gone to someone's head: a young player who wins a few games and starts ignoring his teammates. It carries a hint that the person did not use to be like this, which is why it stings — you are saying they changed for the worse. It is also written swellheaded or swell headed, and Britain has the same idea as big-headed. The related expression 'it went to his head' says the same thing without naming anyone rude. Note that swell here is the verb meaning to get bigger, not the older American slang swell meaning excellent.