How is it that an unchanging and omniscient God can have "regret" as He did in Gen 6:7? In Genesis 6:7, we read, 


  "So the LORD said, “I will wipe from the face of the earth the human race I have created—and with them the animals, the birds and the creatures that move along the ground—for I regret that I have made them.”"


I have never understood how it is that God can experience regret.  Does this not imply that He made a mistake?

Perhaps it is only because as humans we experience regret frequently associated with a mistake; however, regret simply means "to feel sorrow or remorse".  Remorse is defined as (1) deep and painful regret for wrongdoing; compunction.
(2) (Obsolete) pity; compassion. 

I can't shake the sense that regret always entails "I wish I hadn't done that".
God is completely unlike us, so any attempts to describe God using human language are very limited. In this case an attempt is made to express the sorrow of God over the evil and pain introduced into the world, even though he knew it would happen, and he knows he will not only put it right, but make things even better than they were. This passage shows that God is not indifferent toward our problems.