An "alpha" personality is a leadership personality. To lead something, you need to be the best at it. This means that alpha females tend to be extremely competitive. This doesn't necessarily mean they will be competitive in every part of their lives, but they will be competitive in the field they are "alpha" in. This comes from the desire not only to succeed, but to succeed above other people.
An alpha female knows she is good at what she does. This means she can talk to people without fear, can speak publicly and doesn't back down if she thinks she's right. To be in charge, you need to know that you can do it. After all, if you don't, then you can't expect other people to have faith in you. So, alpha females have high levels of self-confidence.
Due to the fact that alpha females are so goal-driven, they can be extremely aggressive. This is because they know what they want and they don't take kindly to people getting in their way. When given the choice between smooth social interactions and achieving their goals, alpha females will choose their goals every time.
An alpha female is self-serving. This doesn't mean she's absolutely selfish, but it does mean that she thinks of herself first. When she makes decisions, she doesn't run through a list of who may be offended and what the possible consequences could be. Rather, she just thinks about whether it's a good decision for her, then does it. She is not overtly selfish, but she does think of what she wants.