When a person is assertive and uncooperative, he is called competing. Competing means that he pursues his needs at the other person's expense. He uses whatever power he has to win an argument or gain the upper hand over the person with whom he has a conflict. For instance, in a conflict between a father and daughter over politics, he may use his position as father to win the argument. The father may say he's right about politics because he is her father. The power position, such as age, status or rank, doesn't have to be connected to the argument; it's just used to defend an argument he believes is correct.
Avoiding style is considered uncooperative and unassertive. The person doesn't participate in the situation. She doesn't seek to satisfy her own concerns or the concerns the other person. In fact, she will use a variety of tactics to not deal with the situation. For example, in a conflict situation with a co-worker, she may withdraw, not saying anything. In another instance, she may sidestep the issue by changing the subject or telling the person she'll discuss the situation later.
Accommodating is a type of cooperative style. The person doesn't take his own needs or concerns into consideration. Instead, it's all about the other person's concerns. For instance, during a conflict situation between a boyfriend and girlfriend, the boyfriend won't avoid the issue but instead will agree to everything being said. He doesn't really agree with his girlfriend, though. He just goes along to satisfy her needs.
There are two conflict styles that involve assertive and cooperative. Collaborating means the person doesn't withdraw from conflict, but seeks to work to find a solution. Typically, she knows there is a problem and is willing to cooperate with the other person to find the point of disagreement. In an example between two co-workers, she will try to find a mutually satisfying solution. The second conflict style is called compromising. This is different from collaborating because the person finds, not a satisfying solution, but common ground.