How do certain versions of the enzyme DNA polymerase know when they made a mistake and how do they reverse?

Well, I don't know the actual mechanism that's used for the polymerase to back up... but I can imagine that the protein can sense a mismatch by the shape of the DNA strand as it passes over it.  If there is a mismatch, obviously the strand will have a "bump" or a shallow spot, depending on the mismatch between base pairs.

This is what wikipedia had to say about error correction with DNA polymerase:
"Error correction is a property of some, but not all, DNA polymerases. This process corrects mistakes in newly-synthesized DNA. When an incorrect base pair is recognized, DNA polymerase reverses its direction by one base pair of DNA. The 3'->5' exonuclease activity of the enzyme allows the incorrect base pair to be excised (this activity is known as proofreading). Following base excision, the polymerase can re-insert the correct base and replication can continue."

I hope this helps at least a little.