Lets say I have two hard drives in RAID 1 and one of them fails, I know that you are supposed to get an identical drive and let the one that didn't fail re-clone itself on the new one.
But my question is that if I have two drives in RAID 1, and one fails, can I just continue running my system as normal using the left over drive as a stand-alone drive in a non-RAID setup or must I have it replaced?
you can keep running it like that but the point of a raid is to have the second (or 3rd or Nth) drive to have a redundant copy in case the other (or one of them) dies too. One failure youre ok... but 2 and youre SOL. Which is better than a stand alone disk with no raid and no external backup... where one drive dies and youre SOL. at least with a raid you have some fault tolerance. So yeah you can run like that but you SHOULD replace it asap in case the other one dies.