Pour some warm vinegar into a plastic sandwich bag, and wrap it around the faucet so that the areas with mineral deposits are submerged in the vinegar. Secure the bag with a rubber band, and allow the faucet to soak for an hour.
You may also notice hard-water deposits on your showerhead. To get rid of them, soak the showerhead in warm vinegar for an hour, then use a toothbrush to remove the loosened deposits. Use a toothpick to dislodge deposits from the spray holes.