One of the first things we do together at Mass is to share our sorrow for the wrongs we've done and the goods we've failed to do, and to ask the Lord for pardon and for peace. Usually we pray the Kyrie: "Lord, have mercy. Christ, have mercy. Lord, have mercy." Sometimes we pray what's called the Confiteor, a prayer that begins, "I confess, to Almighty God, and to you my brothers and sisters, that I have greatly sinned..." (Check out the full text.)
We all screw up from time to time. We all do things that make others' lives harder to bear. Often these actions are mostly accidental. Sometimes they're on purpose, but under the influence of confusing feelings and motivations. Only rarely do we cause others harm out of clear-sighted malice. And as we all know, when we hurt others, if we want to make things right, we must begin by apologizing.
Apologies are gesture of good will. They show the other person that we recognize that our actions were inappropriate, unskilled, harmful. And they show that we acknowledge our own responsibility for what we've done.
Apologies are a first step, but they're only a step. We can use our words to tell people that we regret causing harm, but these words quickly lose their meaning if we don't attempt to repair the damage we've done. Healing a broken relationship takes good intentions, but it also requires time, effort, skillfulness, and the help of others. This is true for human relationships; it is even more true for our relationship with God.