Depending on the situation, make a strategy for their behavior. Let your actions will man realize that you sincerely repent of what is happening and sorry about what happened. The way it should be your apology will depend on the extent of your guilt, and of what it is you are guilty.
Talk to the man. With all sincerity show him what's going on in your soul. Openly tell him how you are going through. If the person you have wronged you is very dear, let him know about it. If people see that you have realized your mistakes and understand what to blame, it will be easier to earn him a full pardon.
It is not necessary to ask the question "How can I make amends? ". It sounds insincere and means that you do not accept his guilt, resentment consider unjustified and unreasonable person and just want to close the topic. So you only spoil everything.
Another way to cause negative emotions and hurt a man even more - it's trying to bribe him with a gift. You will lose their trust.
No need to make any promises. His action or words you have made a sick man. Now, to make amends, you need only to realize it fully accept and draw the right conclusions. Adjust their behavior, you will be able to gradually improve relations with an injured man, and he'll be sure to forgive.