Everyone has it in their heart to forgive. It really depends on what you did, that usually will determine if they will or not. Once you've hurt someone emotionally, you have lost their trust. Getting that back is difficult. If it's something that you continue to do to this person, I can't say they will or even should forgive you. Forgiving starts with yourself and being honest with yourself, then eventually with this person. Have you truly learned what you did is wrong? Do you honestly care how it has effected this person, or just yourself? If forgiveness to you only determines how it effects yourself, then you are just going to continue to do these things that doesn't deserve or should require forgiveness from this person.
Give this person space. When someone gets hurt emotionally, or they are angry, they exaggerate at the time. They will swear they'll never speak to you again, or even say some very mean and hurtful things to you. They are just acting on impulse, from the anger and sadness you've caused them. The best method at first is to give them time. Time to reflect and let their feelings simmer down. If you did something very hurtful, chances are they'll definitely still be upset when you get a chance to talk to them. They might be willing to give you an honest chance again. For now you are probably the last person they really want to be around, so respect that, and leave them alone for awhile.
Examine your own conscience. Okay, if you want someone to forgive you, it starts with yourself first. Simply going up and saying the words "I'm sorry" will not work. They are just words, that they get overused everyday. You say sorry when you accidently bump into someone, or make a simple error. It takes more than those words to ask forgiveness from hurting someone. Take time to really understand what you have done and how your actions effect others. Put yourself exactly in this persons place, and ask how you would feel. Understand where their pain is coming from, not yours. Think about how and why you've done this. Figure out where it came from and how to truly avoid doing this again. Take knowledge over the problem that accorded.
Be willing to actually change from your actions. I think it comes down to this simple manner. People tend to be motivated by their own pleasures and pain. I've learn to recognize this from people in general and chances are you've done this as well, which is why you've hurt this person. But people also make honest mistakes as well. None of us are perfect. Which category do you fall under? Was this a mistake, or did you never take the time to begin with to really think about this persons feelings? Are you honestly going to think about theirs when forgiving? To get someone to forgive you, it starts with really caring about their pain and how you've effected it. Not just simply feeling guilty, to help recover for yourself and your conscience. Feeling guilty over something you've done wrong, is the politically "right thing" to do. It's just a way to tell your conscience it's okay since I feel guilty. Everyone has the ability to tell between right and wrong, and to feel bad. Guilt makes a person feel bad majority of the time for their own self, not truly for someone else's. It's the same reason why people seek revenge over a wrong doing of someone's else's pain. We are motivated by our pleasures and pains. You see the difference? If you aren't truly willing to learn and see how your actions can really hurt someone else, and truly care about it for them and not yourself, you'll do this again. Get over your guilt and just think about the persons pain that you've caused. Take the steps to actually avoid not making the same mistakes again. Selfless people learn, selfish people continue down the same destructive path.
Speak from the heart and be truthful. When the time is right, and you've done these steps. Now it's time to speak to them. If they refuse to talk, give them more time. If they have finally given you a chance to actually be in the same presence, tell them everything. Explain why it is you did what you did. Explain your motives and tell them what you've honestly learned. If you've done the first three steps, this should be easy. You should honestly see things differently and have a better understanding on the pain you caused the person. You should see their pain, as if it were your own. So on a emotional level, you can then get through them. Doing this shows this person you really decided to take the time to honestly reflect on what you have done. You should be completely honest and show emotion, along with passion when you are explaining to this person. Let them talk as well and let them explain how you've hurt them. Listen to what they are saying, because that is important. If you're not listening, you come across as not truly caring. Ask questions on their feelings on the subject, to come across as honestly caring for them.
Do not get angry. If they aren't acting extremely friendly, do no engage in a heated argument. You will have ruin any chance of them forgiving you. Take whatever they give you, good or bad, and be very polite and calm. They might not be over what you've done, but if you act respectful it will help a lot. Let them get the anger out, if it's still bottled up. Be compassionate and show that you truly understand and care. Be very supportive to them, no matter what it is they are saying to you. Don't take it personal, they are hurt.
Don't demand anything from them. If this was an ex-lover, whoever, don't expect or demand that they forgive you. Don't even ask for it. People who have to ask for forgiveness, don't come off as genuine. They don't come off as truly caring. They seem more sad for themselves, rather than the person. If someone truly did something, and the first thing they ask, or want is forgiveness, they don't deserve it. If they truly felt bad, wouldn't the first question, or more important one be how are you? Are you okay? Is there anything I can do for YOU? This should have nothing to do with what you get out of it. If you really care for them, you wouldn't ask or demand anything out of them. You'd just want them to be okay.