Removing hard water residue or stains from your pans and pots is as simple as vinegar and aluminum foil.

Poor some vinegar, any type, into the pot or pan with the hard water residue marks.
Make sure you coat the hard water residue areas with the vinegar. You can do this part with a paper towel.
Let it sit for a while, maybe 5 minutes or so.

Crinkle up a sheet of aluminum foil into a ball.
Using the ball as a scrubber, get it into the vinegar, then scrub wherever there is hard water residue in the pan or pot.

You'll notice the vinegar will begin to turn black. This is supposed to happen.
When you've scrubbed off the hard water residue, dump the dirty vinegar out, and wash the pan or pot as you normally would.

If you find you've missed some of the residue, simply repeat the process until it's clean to your satisfaction.