It's always so hard to admit you may have been wrong! Our human weakness toward pride and self-justification is very strong. But none of us can develop lasting and satisfying relationships without being willing to admit and deal with our wrong doing.
First of all remember that no matter how hard you try, you are NOT PERFECT! Accepting that, makes it a little easier to admit it when you goof-up. And we all do. But don't use that as an excuse for expecting others to accept your wrong doing. "I'm only human" is a phrase for you to use on you when recognizing and facing that "I have done wrong." If you say it to someone else as an expectation that they should accept your wrong doing without a heart felt apology it only sounds like you're blowing off your responsibility for the hurt or other mess you have caused.
So after your own face-off with yourself, now you need to identify how your action or lack of action has effected the other person or even an entire group. Asking forgiveness for what you actually did negatively to them is important. Here are some examples to help you look more deeply and accurately at what you REALLY did wrong:
ACTION
1.Was late to a group responsibility
OFFENSE
1. I wasted your time and energy in waiting for me.
ACTION
2. Failure to do what you said you would
OFFENSE
2. I have chosen to be irresponsible forcing you to change your schedule and lose hope that a project would be completed.
ACTION
3. Spoke disrespectfully
OFFENSE
3. I did not give you the respect you rightfully deserve and assumed something about you before getting the correct facts.
Do you see that the action may seem minor but the hurt and damage caused to others may be quite a bit larger? Owning up to my responsibility takes humility and is very uncomfortable but if I want the relationship to continue in peace, then this is the only way.
There is one more step which helps the other person to let go of the wound you caused. After admitting what damage you did to them, be sure to ask them if they will forgive you. Then wait for their response. If they feel you were not genuine in your apology they may not choose to forgive you. You should respond, "I know my hurt to you may be difficult for you to forgive. I am genuinely sorry for it and am working on making better choices in my life! I hope you will be able to forgive me in the future." Then let it go. You did what you could in humility and grace and now it's their issue to deal with. Romans 12:18 only holds us to the peacemaker standard of "...as much as it depends on you, be at peace with all..." Your last responsibility is then to show love to that person and pray that they would be set free from the Spirit of Bitterness. Heap coals of love on their head because "love never fails..." I Corinthians 13:8. This will embellish your character even if they never change.