Search the English Standard Version of the Bible for the word "forgive" and it shows up in 109 verses, either by itself or as part of the words forgiven and forgiveness. It's obvious therefore that forgiveness is one of the most important concepts in the Bible. Jesus spoke at length about it. Here are some examples:
"For if you forgive others their trespasses, your heavenly Father will also forgive you, but if you do not forgive others their trespasses, neither will your Father forgive your trespasses." (Matthew 6:14-15) "Judge not, and you will not be judged; condemn not, and you will not be condemned; forgive, and you will be forgiven; give, and it will be given to you." (Luke 6:37-38) "Then Peter came up and said to him, "Lord, how often will my brother sin against me, and I forgive him? As many as seven times?" Jesus said to him, "I do not say to you seven times, but seventy times seven." (Matthew 18:21-22) Jesus even asked God to forgive those who nailed him to the cross, saying "Father, forgive them, for they know no what they do." (Luke 23:34)
What can we take from these verses? (1) Simple logic: how can we expect God to forgive us if we're not willing to forgive others? (2) How often should we forgive? Seventy times seven; in other words, always. (3) Notice that the other person doesn't enter into the equation. Jesus says forgive, he doesn't say the other person has to ask for our forgiveness or offer a sincere apology or even be a reasonably nice person. We're supposed to love and forgive, period. As we read in 1 John 4:10-11: "In this is love, not that we have loved God but that he loved us and sent his Son to be the propitiation for our sins. Beloved, if God so loved us, we also ought to love one another." (4) It's not immediately obvious from the verses I quoted, but there's one person we should also forgive: ourselves. This is often the person we struggle most to forgive. If we've learned from a particular mistake we made and are making an honest effort to change our behavior, God is always willing to forgive us. If God is willing to forgive us, then we should be willing to forgive ourselves.
One last point: forgiving someone does not necessarily mean condoning their behavior. You can forgive someone but still say "your actions were inappropriate" or "I can't interact with you if your behavior doesn't change". People need to hear the truth if it's spoken with love and concern. If we say we love our children but we don't instruct them or discipline them, then we don't really love them. Therefore what good does it do to forgive an addict if we don't say "you need to change"? Or how can we say to an alcoholic "I love you" then give them a ride to the liquor store? Yes, we're supposed to love and forgive, but sometimes that means telling someone what they need to hear rather than what they want to hear.