The stains are likely deposits of salts that were leached from the soil in the pots. The salts come from the fertilizer in the soil. I don't know if these will work, but they're worth trying. Sometimes vinegar will remove salt stains. It takes old-fashioned elbow grease and a good stiff brush. Give that a whirl. If it doesn't work, try oven cleaner. Spray it, let it sit 10-15 minutes, then flush it away with a pressure spray from your garden hose. Another cleaner option is that oxy-clean stuff for laundry (or some similar product). Mix 1 fluid oz. of that per quart of water (4 oz. per gallon) and scrub the rings with that. I have the most faith in that or the oven cleaner for removing your stains.
This year, flower pots on the cement are fine, just put some plastic saucers under them. You can elevate those saucers using pot feet to avoid trapping any moisture underneath them, if you feel you need to. They sell little plastic pot feet and terra cotta ones (pricier). You usually need 3 per pot, 4 for larger pots.
When you buy saucers for your pots, try to find ones that are are flat as possible. I've noticed that some of the saucers have ridges in them, perhaps to lift the plant out of sitting in water. This creates a situation of standing water that must be dumped. I try to find saucers that are flat and when I do, I buy a bunch in assorted colors to match my pots. You can always buy the cheapest ones they make, clear plastic, to go with your pots.
Have a wonderful gardening season!