Cloudy Mirror / Celestial Mirror:

In your travels around the world of Spira, you may come across chests you can't open, and mysterious mandalas on walls. These chests and mandalas house your characters' ultimate weapons, and require a charged magic mirror to access.

To start, head to the Calm Lands and obtain a chocobo from the stationary (ie, not moving) chocobo rider there. Ride to the south plateau near where you came in from Macalania Woods, and on the east side you should see a broken bridge and a chocobo feather on the ground. Press [x] near the chocobo feather, and your chocobo will fly to the plateau below it. Continue east to the Remiem Temple. Examine the sphere on the left side of the temple, then talk to the chocobo on the right to race the second chocobo on the left. If you make it to the center of the circular track quickly enough, you'll be rewarded with the Cloudy Mirror.

Side note: if you keep racing here, you can acquire other fine and rare
items, depending on how many treasure chests you were able to reach while
still winning the race. These prizes are:

0 Treasure Chests: Potion
1 Treasure Chest : Elixir
2 Treasure Chests: Mega Elixir
3 Treasure Chests: Wings to Discovery x30
4 Treasure Chests: Pendulum x30
5 Treasure Chests: Three Stars x60

Do NOT touch any poles while racing, or you'll end up with a lousy potion instead of the above listed rare prizes. (Touching poles will not prevent you from getting the Cloudy Mirror, however.) I'm pretty sure the prizes are one-time-only deals, so don't kill yourself trying to obtain Wings to Discovery over and over.

To de-fog and charge your newly acquired magic mirror, talk to the mother and son at the entrance of Macalania Woods. Find the father at the campsite, speak to him, and he'll run off to be reunited with his family. Go back to theentrance and speak with the mother and father again, and keep talking to them until they mention their missing son. This may take 3 or 4 tries. After they mention their son, head up the sparkly path to their left into the trees. Bear north at the first fork, where the son should allow you passage if you talked to his parents enough times. Show the Cloudy Mirror to the large ambrosia salad-looking plant, who'll turn it into the Celestial Mirror. If there's a woman blocking the path instead of the son, either a) you're not far enough in the game yet, or b) you didn't talk to the parents enough times. With the Celestial Mirror, you can now open chests and detonate the mandalas protecting your ultimate weapons.