AGGRESSIVE - Those who are more interested in having their own needs met than respecting others use aggressive techniques to get what they want. They may be demanding, loud, and even abusive in their efforts that tell the world "I am important and you need to serve me".
PASSIVE - When an individual is afraid of being aggressive or thinks that his/her needs are not important, that person might ignore or not talk about what they need or want and defer to the aggressive individuals in their lives. They might stay in abusive relationships for long periods of time. Their self-image is low and they might not even look others directly in the eye. Unfortunately, by repressing their needs and ideas, they often end up becoming very resentful or ill.
PASSIVE-AGGRESSIVE - Some people do not like others but instead of confronting them, they choose to do things that are indirect - such as gossiping about the person or trying to turn others against the target. The example that I use with clients about this style is of the chef who doesn't like you but, instead of learning to resolve issues, s/he just spits in your soup!
ASSERTIVE - When you know what you need and are able to ask for it in a healthy manner, you are an assertive person. You do not need to go into long explanations to express your opinion and are clear in your communication styles. The result is that you do not avoid issues or feel resentment but are able to deal with things right away.
Let's use an example. Suppose that another professional asks to borrow your laptop. The aggressive style might be rude and say something that is inappropriate such as "You've got to be kidding. I am not here to make your dreams come true! Get your own laptop!".
The passive person might lend the computer even though they have projects that need to be done. Then, after the professional leaves with the laptop resentment festers. (This person likely will not ask for the laptop to be returned early or will do so in an aggressive manner that shocked the borrower).
The passive-aggressive person might agree to lend the laptop but not give the borrower the power cord.
The assertive person would say "I am not able to lend this to you because I have projects that need to be completed today".
You see, passive desperately wants to be liked. Aggressive doesn't care what others think and assertive focuses on issues rather than the state of the relationship.