Having called for a righteousness which exceeds that of the Pharisees, Jesus provides six examples (Matthew 5:21-48). He takes a generally accepted version of a law or saying, then insists that his followers go farther and deal seriously with the sources and consequences of actions. The first of the six examples is Matthew 5:21-26:
“You have heard that it was said to those of ancient times, ‘You shall not murder’; and ‘whoever murders shall be liable to judgment.’ But I say to you that if you are angry with a brother or sister, you will be liable to judgment; and if you insult a brother or sister, you will be liable to the council; and if you say, ‘You fool,’ you will be liable to the hell of fire. So when you are offering your gift at the altar, if you remember that your brother or sister has something against you, leave your gift there before the altar and go; first be reconciled to your brother or sister, and then come and offer your gift. Come to terms quickly with your accuser while you are on the way to court with him, or your accuser may hand you over to the judge, and the judge to the guard, and you will be thrown into prison. Truly I tell you, you will never get out until you have paid the last penny. (Matthew 5:21-26)
Like any teacher of his day, Jesus accepts the prohibition against the act of murder. He goes on, though, to meddle with the human heart. Anger, says Jesus, is the breeding ground for murder. It can easily lead one to lash out with words, such as "fool" or "idiot. Left to fester, anger can incite murder.
Note that Jesus does not forbid the feeling of anger. He knows very well that feelings simply happen. Instead, he teaches that his followers are to face the reality of the feeling, step away from whatever task is at hand (even the act of worship) and go seek reconciliation with one's brother (or sister, for that matter).
Jesus extends the same principle to a less personal situation: if someone is taking you to court, deal with your anger by seeking reconciliation with that person.
Insofar as I can tell, Jesus is calling for behavior consistent with the kingdom of heaven. One such indicator is a passion or commitment to reconciliation, to restored relationships. Interestingly enough, Jesus does not promise the other person (the brother, the adversary, etc) will respond in kind. His followers tend to chafe a bit at this point. For example, Peter will later ask how many times he must forgive someone who offends him. Jesus' classic answer is,"Seventy times seven," a proverbial saying which means "without limit."