There really is no single product that will work on all shower doors. Your best bet is to try different things on small areas of the shower door to see what works best.
First thing to know is whether you have glass shower doors or plexiglass shower doors. If you use the wrong product, you can ruin your shower doors.
Assuming you have glass shower doors, here are some things you can try:
1) Use Jet-Dry and a Magic Eraser sponge. Jet-Dry is a product normally used in dishwashers to keep glasses from developing hard water stains, and it actually works well on shower doors.
2) Use a Magic Eraser sponge with no cleaning product, just use it dry. Scrub the shower doors in small, circular movements.
3) Use a drywall sponge with dish soap or Jet-Dry. Drywall sponges look like thick sponges wrapped in sandpaper and can be found at hardware stores.
4) Use a razor blade and scrape it off.
Whatever you use, once you get it clean, I highly recommend that you use some Rain-X on the shower doors if they are glass doors. The Rain-X will cause the water to bead off and help prevent future problems with hardwater deposits. You can find Rain-X at auto supply stores.
Good luck! Please feel free to contact me with any other questions: firstname.lastname@example.org.