So if you are offering your gift at the altar and there remember that your brother has something against you, leave your gift there before the altar and go. First be reconciled to your brother, and then come and offer your gift. (Matt 5:23-24)
The problem is that this is easier said than done. How are you supposed to do this?
The answer to this question lies in the context of Jesus’ words — the Sermon on the Mount. Here Jesus is laying out a completely different way of life, one that many (or most) people struggle to attain. For instance, rather than treating people the way they treat you, you allow them to mistreat you (Matt 5:39-41), and to perform radical surgery on your life to avoid sin (Matt 5:2-30), and to really trust in God (Matt 6:24-25). Taken as a whole, it’s radical prescription for a totally different way of life, focused on meekness and humility. Jesus himself recognizes that this is hard:
Enter by the narrow gate. For the gate is wide and the way is easy that leads to destruction, and those who enter by it are many. For the gate is narrow and the way is hard that leads to life, and those who find it are few. (Matt 7:13-14)
So, back to the question — how to be reconciled? It’s clear that Jesus’ intent is that you go and apologize, perform restitution, etc, to the degree that you are able (which is probably a lot more than the degree to which you are comfortable).
Blessed are the poor in spirit, for theirs is the kingdom of heaven.
Blessed are those who mourn, for they shall be comforted.
Blessed are the meek, for they shall inherit the earth. (Matt 5:3-5)