5) This is the part that most people forget to do. This when it is important to explain what the person means to you, and how much you want to make amends. This makes any apology go above and beyond, to the point of actually helping to move towards reconciliation. You wouldn't apologize if you didn't want the other person to accept your apology, so make sure they know what your relationship (working, friendship or otherwise) means to you, to let them know that it has some value to you and is worth fixing. If appropriate, this can be a good time to give the person a compliment (that is in some way connected to the precipitating incident) and explain the importance they have in your life; you have hurt this person, it's time to try to make them feel better.
6) Repeat these steps until your apology is accepted and you are able to make peace with the person.
7) Be open to letting them say what hurt/bothered them. Even if they've already told you before. Sometimes they will need to say again why they are upset. PAY ATTENTION TO WHAT THEY SAY! This is what you must mention again in steps two and three. If they feel the need to repeat it, it is probably because they do not feel that you have heard them/understood. If, after repeating step two twice or more, ask them if you have covered everything or left anything out. Remember, you want to make sure they know that they've been heard, and sometimes just checking will make them know that this is important to you and that you really want to make sure you understand why they are upset.
8) At the beginning of the next conversation, it is helpful to just make sure the person is feeling okay. Sometimes, new things bubble up after an apology, and if you want to move forward in peace with that person, just make sure that everything is still as you left it at the end of your apologetic conversation.
Of course, every person has their own style, and specific things that may be more or less important to them. However, if you follow these steps, your apology is more likely to be successful. Now remember, if it is insincere, the other person will know! These steps will only work if you truly feel remorse, and are just unsure of how to express it.