According to wikiHow:
We all know what an apology is; it's an expression of remorse or guilt over having said or done something that is acknowledged to be hurtful or damaging, and a request for forgiveness. But we also know it can be really hard to swallow our pride and say "I'm sorry." If you have a difficult time making amends for mistakes or repairing the effects of angry words, here's how to keep your dignity while being humble, and invite forgiveness with grace.
1. Determine what went wrong. Did you say something insensitive, no matter how true it is? Did you fail to come through on a promise? Was the offense recent or long ago? You can't apologize effectively if you don't know what you are apologizing for. If you don't think you did anything wrong, then express regret or sadness for the feeling that someone is experiencing as a result of what you did. Presuming the effect was unintended, the basis of the apology often lies in not having foreseen how your actions would affect this person, realizing that the benefits of the action did not outweigh the unforeseen circumstances, and wanting to compensate for your oversight.
2. Take full responsibility for the offense, without sharing the blame with anyone else, and without presenting mitigating circumstances. Admit that you were wrong emphatically, unreservedly, and immediately. An incomplete apology often feels more like an insult. An apology with an excuse is simply not an apology. It may very well be that other people or circumstances contributed to the situation, but you cannot apologize for them; you can only apologize for yourself, so leave them out of it.
3. Realize that there are no excuses. Do not try to think of or offer one. An apology with an excuse is not an apology. Take full responsibility for what you did. And if the person to whom you apologize rejects it, then they do not deserve it, but do not take it back; still say "I'm sorry".
4. Decide when to apologize. Sometimes immediately after your mistake is best, sometimes not. The sting of a harsh word can be cooled right away with a quick apology, but other offenses might need the other person to cool down before they are willing to even listen to your next sentence. However, the sooner you apologize for your mistake, the more likely it will be viewed as an error in judgment and not a character flaw.
5. Write your apology down. Construct a letter to the person you're apologizing to, rehearsing what you will say in person. If you don't feel comfortable with writing, then use a voice recorder. Not only will this help you remember what to say when you're face to face with them, but you can also bring the copy with you and hand it to them if you find the apology quite difficult to express. But never forget that a direct and honest apology is best. Do it face to face, if possible. A phoned, emailed or recorded apology may show a lack of sincerity and effort.
6. Begin the apology by naming the offense and the feelings it may have caused. Be specific about the incident so that they know exactly what you're apologizing for. Make it a point to avoid using the word "but". ("I am sorry, but..." means "I am not sorry.") Also, do not say "I'm sorry you feel that way" or "I'm sorry if you were offended." Be sorry for what you did! "I'm sorry you feel that way" makes it seem like you are blaming the other person, and is not a real apology. Validate their feelings or discomfort by acknowledging your transgression's (potential) effects, while taking responsiblity:
* "Boss, I'm sorry I'm late again, I know my shift started 10 minutes ago. I hope this doesn't complicate your day."
* "Dear, I'm sorry I forgot your birthday - there's no excuse. I hope you don't feel neglected, please let me set this right."
7. Make amends. Think about what caused you to make the offense. Is it because you're a little too laid back about being on time, or remembering important dates? Is it because you tend to react instantly to certain comments, without pausing to consider an alternative point of view? Is it because you are unhappy with your life, and you unknowingly take it out on others? Find the underlying problem, describe it to the person (as an explanation, not an excuse), and tell them what you intend to do to rectify that problem so that you can avoid this mistake in the future:
* "I snapped at you because I've been so stressed out with work lately, and it's selfish of me to take it out on you. Starting tomorrow, I'm going to cut down my hours to X per week. I really think it'll help me unwind, and help us spend more quality time together."
* "I've been distant and cold because I get paranoid that you're going to walk out on me because I don't have a job. But that's a terrible thing to do. Look, here's a list of things I'm going to do to find a job ASAP..."
8. Express your appreciation for the role that they play in your life, emphasizing that you do not want to jeopardize or damage the relationship. This is the time to briefly recount what has created and sustained the bond over time and tell loved ones that they are indeed loved. Describe what your life would be missing without their trust and their company.
9. Ask if they will give you another chance to make up for what you did wrong. Tell them you'd love to show them that you've learned from your mistake, and that you will take action to change and grow as a result, if they will let you. Make a clear request for forgiveness and wait for their answer. This gives the injured party the well deserved "power" in determining the outcome of the situation.
10. Be patient. If an apology is not accepted, thank them for hearing you out and leave the door open for if they wish to reconcile later. (E.g. "I understand you're still upset about it, but thanks for giving me the chance to apologize. If you ever change your mind, please give me a call.") Sometimes people want to forgive you, but they still need a little time to cool off. If you are lucky enough for your apology to be accepted:
* Avoid the temptation to throw in a few excuses at the end. Instead, have a transition planned out beforehand for what you can do to solidify the clean slate (e.g. "Let's go get some coffee and catch up. It'll be my treat. I miss knowing what you're up to.").
* Remember, just because someone accepts your apology doesn't mean they've fully forgiven you. It can take time, maybe a long time, before the injured party can completely let go and fully trust you again. There is little you can do to speed this process up, but there are endless ways to bog it down. If the person is truly important to you, it's worth it to give them the time and space they need to heal. Don't expect them to go right back to acting normally immediately.
o At the same time, don't let someone hang this over your head for the rest of your life. The same way you need to learn how to apologize, they need to learn How to Forgive.
11. Stick to your word. This is every bit as important as every other step. A true apology entails a resolution, and you have to carry out your promise in order for the apology to be sincere and complete. Otherwise, your apologies will lose their meaning, and trust may disappear beyond the point of no return.