When people sin against us but don't repent, should we forgive and forget? Someone sins against us, e.g. says something unkind. They are completely unrepentant. That might be because they don't think it is a sin or they don't care (yes, I'm deliberately mentioning two different attitudes).

Should we forgive?

And should we forget?

I know Jesus forgave on the cross, saying, "they don't know what they are doing." Was this attitude something we should always try to copy?
Depends, are they a believer or an unbeliever?
Unbeliever
(Mark 11:25 ESV)
&quot;And whenever you stand praying, forgive, if you have anything against anyone, so that your Father also who is in heaven may forgive you your trespasses.”
When you forgive an unbeliever you are acknowledging that they have wronged you but you will not try to seek justice yourself, you will leave that to God. In this way you may live peacefully not worrying about getting tit-for-tat in a never ending cycle with the other person.
Believer
(Luke 17:3, ASV)
&quot;Take heed to yourselves: if thy brother sin, rebuke him; and if he repent, forgive him.&quot;
In the first part of this verse Jesus issues a warning, what follows is important. Next comes two If/Then's which start out addressing the cause if your brother sins. Most people would agree that 'brother' here means fellow believer. So if a fellow believer sins your first action is to rebuke him not forgive him. From here there are two options. Either he repents and you forgive him (the second If/Then) or he doesn't repent. In the second cause we can look to the Gospel of Matthew.
(Matthew 18:15-17, ASV)
&quot;And if thy brother sin against thee, go, show him his fault between thee and him alone: if he hear thee, thou hast gained thy brother. But if he hear thee not, take with thee one or two more, that at the mouth of two witnesses or three every word may be established. And if he refuse to hear them, tell it unto the church: and if he refuse to hear the church also, let him be unto thee as the Gentile and the publican.&quot;
Once again this verse is addressing what to do when a brother (fellow believer) sins against you. If he hears you, not necessarily agrees with you, he is still your brother. Otherwise if he doesn't hear you then you gather witnesses. If he still doesn't hear you then he should not be in community with you like another believer since the both of you would not be able to live in harmony. And @ChoasGamer has some verses that also point that out.
Now Jesus didn't say that he must repent first but it is in your best interest to not forgive him till he repents otherwise you pass judgement on him.

Should we forget?

Not necessarily. If someone sins against you, especially someone close to you, it will take time for them to gain your trust and most people would agree that this is healthy. Also trying to forget something you know to be true can be unhealthy.  Forgiveness is stating your will that you will not try to get justice back from the other person it doesn't necessarily deal with your emotional response.
This is just a brief answer on the topic. For more information check out this site which I used to form this answer.
The main difference between this approach and the answer by Sonic is the Biblical consideration of NOT ignoring the person. 1st Corinthians 5:10 shows that we remove the unbeliever from community SO THAT WE CAN fellowship with them in order to win them back to Christ!