I will remember their sins no more (Heb 8:12; 10:17; cf. Isa 43:25).
It's striking how often Bible scholars and theologians talk about forgiveness without beginning the analysis with a definition. Perhaps they think the meaning of forgiveness is self-evident. Or perhaps they think the concept is so basic that there's nothing more fundamental they can compare it to.
Still, given the centrality of forgiveness in Christian theology, you'd think there'd be more effort to define the key concept.
On rare occasion, Scripture uses the anthropomorphic notion of divine forgetfulness to illustrate the nature of forgiveness. That's a useful analogy, because it's a unifying principle.
i) It supplies a conceptual common denominator for many different kinds of Biblical statements that mention forgiveness. The basic idea is that forgiveness is like forgetfulness. When God forgives someone, it's as if God forgot what they did. It wipes the slate clean.
ii) Likewise, it supplies a conceptual common denominator for divine and human forgiveness. For the same principle is applicable to divine and human relations.
iii) Finally, it covers both the psychological and the activist dimensions of forgiveness.
If you forget what someone did to you, then you can't harbor resentment for what they did. When you look at them, it doesn't remind you of what they did.
Likewise, if you forget what someone did to you, then you don't take punitive or vindictive action.
The point is not that God literally forgets our sins, or that we necessarily forget wrongdoing. Rather, forgiveness has the same effect as forgetfulness in that regard. In fact, it's all the more impressive to treat someone as if you forget their misdeed, even though you remember it.
iv) There are, of course, related concepts in Scripture. Justification is a richer concept than forgiveness. Forgiveness means treating the offender as if he's innocent, whereas justification means treating the offender as if he's good. It's a positive ascription. Not merely the absence of guilt, not bare acquittal, but a positive status.