I recently wrote a how-to on here that gave a step-by-step guide on how to apologize. Yet this has still left a void, I realize, so here is the counterpart to the apologizing instructions.
So, you've had a conflict. Whether you've already done your apologizing, or you are the rare "completely-wronged-I-didn't-do-anything" party, you are about to receive an apology. But it's just not that simple. If a person apologizes correctly, they will be showing their remorse, and a simple, "okay" is often not a proper acceptance. So here are some simple instructions to help you and the apologizer move forward in peace.
Preparation:
1) Get your concerns/issues/problems/complaints straight in your own head, first. After a fight, the real causes get lost, so it is important to really know why you were upset and/or hurt. This links with step 2.
2) Figure out what would make you feel better. This is VERY IMPORTANT! You must know what kind of apology you want, and how you want to move forward. Otherwise, if you are not satisfied with the apology you receive, you will not exactly know why, and not be able to voice your concerns or what would make the situation better. These need to be concrete actions/words; vague statements such as "let me know you care" or "leave me alone" or similar demands are difficult to fulfill and vary from person to person. Therefore, you must clarify for yourself what you mean with these ideas, and express them clearly to the other party.
3) Try to find your own fault in the situation. Are you setting the other person up for failure? Are your demands reasonable? Try to see what you are doing to help or hinder the communication, and its breakdown, and change this behavior. These could also be good things to bring up when receiving the apology, as this avoids the "I'm the good one, you're the bad one" element that makes many people unwilling to apologize.
Receiving the apology:
1) LISTEN TO THE OTHER PERSON! Often, people are not satisfied with an apology, because they do not listen to it. The other person is trying to make amends, give them a shot.