To be proud is to be pleased with someone's qualities or accomplishments. You can be proud of yourself or of someone close to you. Proud has a more positive feeling to it than vain, but it can also be used in a negative way.
Ex: Leo's father, who had wanted to be a famous athlete himself, was extremely proud of his son because he was such a good footballer that he was made team captain.
Ex: Margarita is an unbearable proud woman; I have often noticed that she will not even bother to speak to anyone who is below her in society.
To do one proud means to be a source of pride to someone. Or it can mean to treat someone generously.
Ex: Catherine, you did us proud with that meal. I haven't had a roast chicken like that in years, and the pie was fantastic. You must have spent a lot of time cooking today.
Big¬headed means you have a lot of pride. The idea is that something about you has made you so pleased with yourself that your head has grown to an unusual size.
Sometimes people will tell others not to let something go to their head. This is the same idea as being big-headed. If you think too much of yourself or your accomplishments, it will go to your head and make you think yourself better than others.
Ex: Sarah is at the top of all her classes in school and all her teachers have great respect and admiration for her, but she tries not to let that go to her head.