"forgive and forget" is an expression meaning to completely absolve someone of responsibility for a wrongdoing against you. The idea is that you have not only decided not to hold it against them; you've decided to live as if it had never happened. It's not really a recommendation for behavior, as many such sayings are. People who forgive and forget are seen as both extremely good-hearted and somewhat gullible.

One reason the expression is useful is because it permits a classification system. For example, say we're at a party and I throw up into your favorite top hat. You can:

1) neither forgive nor forget: Come to the next party and continuously guilt-trip me about it
2) not forgive, but forget: Come to the next party and act strangely cold towards me, for no apparent reason
3) forgive but not forget: Come to the next party and continue to be my friend, but refrain from wearing hats around me.
4) forgive and forget: Come to the next party, engage in friendly conversation with me, and show off your new $2000 art object hat made out of air sickness bags without fear or reservation.

I generall consider forgiving but not forgetting to be the wisest and most compassionate way to go.