After considering our physical needs, Jesus teaches us to present our spiritual needs to God. This begins with a request for the forgiveness of our sins (Matt. 6:12; Luke 11:4).
To forgive means "to send away." Sin separates us from our fellowship with God (Is. 59:1-2; 1 John 1:5-7). Even the most devout Christians will continue to sin (1 John 1:1:8-10). The consequences of this separation from God is spiritual death, which will result in eternal damnation. Thus, it is very important that our sins be forgiven or "sent away."
While Luke's account teaches us to ask for the forgiveness of our sins, Matthew's gospel reads, "And forgive us our debts, as we forgive our debtors." Sin is more than a violation of God's law; it is a violation of God Himself. Such an offense places us in debt to God - a debt which we cannot begin to repay. This is illustrated very clearly in the Parable of the Unforgiving Servant, in which a servant was forgiven by his master of a debt he had no ability to repay (Matt. 18:23-35). Instead, he appealed to his master for mercy, and was forgiven the entire debt.
This teaching on prayer also indicates an important condition we must meet if we are to receive the forgiveness of our sins. We must be willing to forgive others of the sins they have committed against us. The importance of this condition is indicated in the verses immediately following the prayer. "For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you do not forgive men their trespasses, neither will your Father forgive your trespasses" (Matt. 6:14-15). It is also the main point of the Parable of the Unforgiving Servant (Matt. 18:35).
There is no point in asking God to do for us what we are unwilling to do ourselves. We must have the forgiveness of our sins, so we must be willing to forgive others of their sins against us.