you have a conflict with someone Scripture calls upon you to do what you can to
resolve it. This can happen immediately  after the conflict (best case
scenario)or at any point following. No matter when it happened, it has to be
you must get the log out of your own eye (Matt. 7:1-5).  Examine yourself
and your conduct in the conflict. Were you unkind, offensive, or cruel? Did you
intentionally set out to hurt or malign the person? Ask the Lord to search your
heart (Ps. 139:23) and reveal to you what your part in the conflict or
disagreement was.  Once you see your part, then you confess it to God as
you confess it to God, then you must repent of it. Repentance is more than “I am sorry” or
“I apologize” it involves several distinct spiritual and actual
changes. When you repent of your sin you see it for what it is and determine to
turn away from it because it doesn’t glorify God. 
you confess and repent you go to the person you have conflict with. Confess
your sin to them and ask them to forgive you.  At this point the
relationship should be restored. 
what if it isn’t sin? What if it is a misunderstanding or the person is simply
angry at you because you refused them something? Nonetheless they are hurt and
angry with you.  Can you just let it go? Can you leave such a fracture in
your relationship with them? 
have been in this position before and it is not easy whether you are dealing
with a fellow Christian or an unbeliever. In such a case explain the situation
to the best of your ability, be sure you tell them you meant no offense or harm
and ask them to forgive you.  
do you do when the person you offended (intentionally or unintentionally)
refuses to forgive you? Unfortunately, I have been in this predicament too.
 You are responsible before God to clean up your end of the mess and
honestly, that is all you can do.
 You are not responsible for the other person’s unforgiving heart or for
any bitterness that grows there. 
surely is a difficult situation because I think the nature of a Christian is to
be a peacemaker and a reconciler.  Our desire is to reflect the changes
Christ has made in us when He forgave us of our sin.  Personally I am
brokenhearted when someone refuses to forgive me, especially when I have done
them no intentional wrong or harm. The only person I am responsible for is me,
and the same is true for you. 
you have to accept that relationship is broken. It may be for a time and it may
be forever.  Keep showing that person the love of Christ despite how they
may treat you. This is what honors God.