A person with a "big ego" believes that he holds significant importance in the lives of those around him and the world in general. While it is often unkind to say someone has a big ego, people who achieve greatness in their endeavors often have one.
Having a big ego is the opposite of having humility. A person with a big ego may have a "can-do" attitude, which helps him fight through challenges to achieve goals. A big ego can also cause someone to get defensive when criticized and to compare himself and his abilities to those of friends or co-workers.