Make eye contact. Using a sincere voice, say:
Be specific. Show the person you're apologizing to that you really understand what they are upset about.
Wrong: I'm sorry for being mean.
Right: I'm sorry for saying that nobody wants to be your friend.
This might take some more thinking, but this is one of the most important parts. Until you understand why it was wrong or how it hurt someone's feelings, it's unlikely you will change. This is also important to show the person you hurt that you really understand how they feel. Sometimes, people want to feel understood more than they want an apology. Sometimes just showing understanding — even without an apology — is enough to make them feel better.
Wrong: This is wrong because I got in trouble.
Right: This is wrong because it hurt your feelings and made you feel bad about yourself.
Use positive language, and tell me what you will do, not what you won't do.
Wrong: In the future, I will not say that.
Right: In the future, I will keep unkind words in my head.
Wrong: In the future, I won't cut.
Right: In the future, I will go to the back of the line.
Wrong: In the future, I won't push.
Right: In the future, I will keep my hands to myself.
Wrong: In the future, I won't take your eraser.
Right: In the future, I will ask you if I can borrow your eraser.
This is important to try to restore your friendship. There is no rule that the other person has to forgive you. Sometimes, they won't. That's their decision. Hopefully, you will all try to be the kind of friends who will forgive easily, but that's not something you automatically get just because you apologized. But you should at least ask for it.