FFX - Celestial Weapons

In Final Fantasy X, you need to collect the weapon itself, and a crest, and a sigil.  You'll then need to treat them with the Celestial Mirror to actually make them useful.

Celestial Mirror

You'll win the Cloudy Mirror in the Chocobo Race at Remiem Temple, for finishing first, with no treasure chests opened.

Take it into Macalania Woods, and speak to the mother and child.  Find the father near where you camped, and then talk to the couple.  You'll find the son on the crystal path which goes above the woods.  When you examine the crystal he was looking at, the Cloudy Mirror will become the Celestial Mirror.  You'll need to come back here when you've collected everything.

Tidus

Weapon - In the Calm Lands, beat the chocobo trainer's time to pull the guard away from the crevice in the north-west.  You can grab the sword if you're in possession of the mirror.  Don't try it before then, or you'll have the beat the stupid game twice.

Sigil - Beat the stupid Catcher Chocobo game with a total time under 0:00 seconds.

Sun Crest - In the Zanarkand Dome.

Yuna

Weapon - capture one of every monster in the Calm Lands and you'll get it from the owner of the Arena.

Sigil - Defeat all of Belgemine's aeons and send her to the Farplane.

Crest - On Besaid Beach, in a little alcove to the right, away from the main beach.

Auron

Weapon - you'll find the rusty sword near the entrance to the sunken cave.  Take it to the statue of Lord Mi'ihen on Mushroom Rock Road.

Sigil - Generate ten creatures in the monster arena.

Crest - Oldroad South.

Wakka

Weapon - from the owner of the cafe in Luca after either 100 Blitzball victories or placing higher than third in a league or tournament.

Sigil - Offered as a Blitzball prize after winning the Attack Reels, Status Reels, and Aurochs reels.  You'll need to have fought a total of 450 battles by this point.

Crest - In the locker room at Luca once you have the airship.

Kimahri

Weapon - activate three Qactuar stones in the Thunder Plains, then find the Qactuar ghost in the south.  Follow it until you get the chest containing the Spirit Lance.

Sigil - Beat the Butterfly hunt after obtaining the airship.

Crest - Mt Gagazet.

Lulu

Weapon - Baaj Temple after defeating Geosgaeno.

Sigil - Dodge 200 lightening strikes.

Crest - on the Farplane, once you have the Airship.

Rikku

Weapon - enter the password GODHAND on the airship to find it.

Sigil - complete the Valley of the Catuars minigame.

Crest - Sanubia Desert.