Understand how serious the problem is
Understanding the problem that has driven a wedge between you and your friend is the key to sorting things out. Try to assess what went wrong and what caused you to fall out with him/her. Sit alone for a while and assess who was at fault and who went overboard. Being alone for a few hours will help you know who was mistaken and who acted sensibly.
Kill your ego
Unless you kill your ego, you will never be able to sort things out between you and your friend. Ego often aggravates the problem because it will keep you from making up with your friend. No matter who is wrong, ego cane causes trouble to both of you. If you think you are right and your friend too realizes that he/she is mistaken, things will remain the same between you two if you let your ego control your emotions.
Do not let your ego further hurt your relations with your friend and be the first to shake hands.
Be brave to forgive if your friend is at fault
You should be brave enough to forgive. Being forgiving always works and it makes things seem good once again. If your friend has made a mistake or hurt you, and to make matters worse, he/she is rigid enough to admit his/her fault, it is time for you to take an action and put your relations on track.
Forgive your friend by contacting him in person. If you feel a bit awkward, you send him a hand-written letter saying that you did not take offence at what he/she did.
Apologize if you are at fault
If you know that you are mistaken and have done something wrong, do not expect your friend to come to you and shake hand or hug. You are the one that caused trouble, so you have to apologize. If you are shy and cannot face him/her, write an email or send a message through mobile phone.