Whenever you hear "I would have ___", you can always imagine that the sentence ends with "...if ___". For example:
I would have remembered.
I would have remembered even if you hadn't emailed me.
"I would have ___" is used to talk about something that didn't really happen, but you're imagining how things would be different if something different had happened in the past (like if your sister hadn't reminded you of your mom's birthday).
It doesn't matter whether the action would have happened in the past or in the future. You use "would have (done)" to talk about something that's about to happen:
A: Don't forget, we have a meeting this afternoon at two.
B: Don't worry, I would have remembered!
But you would also say the same thing if your sister mentions that she remind you of your father's birthday six months ago:
I would have remembered that too!
A note on pronunciation: "would have" is usually pronounced "would've", which might also sound like "would of". Some English speakers who don't write very well even make the mistake of writing "would have" as "would of".