Most people consider themselves one or the other, and some consider themselves to be ambidextrous.
To be ambidextrous is to be able to do things equally well with either hand. Very few people are truly ambidextrous. The most common way to determine if one is left handed or right handed is by which hand the person writes with. Probably the second most common is the hand used for throwing a ball. Other things, such as how one swings a baseball bat or a golf club, are often irrelevant.