Celestial Weapons are the best, or ultimate, weapons of Final Fantasy X.
Every Celestial Weapon has four abilities on them, all of which will have Break Damage Limit. And even though it's not listed as an ability, they will have the Piercing effect as well.
When you first obtain these weapons you may first think, 'Wow what a useless weapon, No AP!'. Unfortunately, simply acquiring these weapons is not good enough, as they will all start off with only one ability, No AP. To unlock the true powers of these weapons, you must obtain two items for each of them, Crests and Sigils.
Now, in order to find these weapons, you must first obtain the Celestial Mirror.
 How To Obtain The Celestial Mirror
In order to get the Celestial Mirror, you first need to obtain the Cloudy Mirror. You first must travel to Remiem Temple. To get there you need to be able to ride a chocobo in the Calms Lands. To do this, talk to the Chocobo Trainer found either northwest in front of the Travel Inn or on the ledge to the right when you enter the Calm Lands from the South. (Note that when you enter the Calm Lands for the very first time the Trainer will only be found northwest). Then you must complete her chocobo training session (see Chocobo Trainer). When you are able to ride one, do so and take it to the south entrance and go right. You will see a broken bridge and a chocobo feather. Press the X button to examine the feather and you will jump down. Then follow that path which leads you to the bridge before Remiem Temple.
At the entrance to the Temple take a left and press the X button to examine the sphere on the ground. It will just tell you about chocobo racing. Go around to the other side and talk to the chocobo and say you would like to race, which is simply a run all the way down to the bottom (don't worry about the chests or the poles). If you make it to the bottom first (which is simple unless you were mucking around), you will win the Cloudy Mirror, the first piece to the puzzle.
Now you must get the Celestial Mirror. To do this you must go to Macalania Woods. On the south side of the Woods (where the entrance to Thunder Plains is), you will see a woman and a child. Talk to both of them and they will say that they are waiting for their husband/dad. Take the path on the right, carry on going right till you're at the crossroads. Take the north path and talk to the man there and tell him his family is waiting for him. He will leave, so follow him to his family, but this time the son is missing. Talk to the mother and father and then take the crystallized path on the left and follow it until you find a path that will take you north. A short distance up that path is the son. Talk to him and the Cloudy Mirror will start to react, prompting you to use it. Congrats, you now have the Celestial Mirror, the second piece to the puzzle.
This sphere that has transformed your Cloudy Mirror into the Celestial Mirror is very significant, as you will be returning here to unlock your Celestial Weapons' true abilities.
Below is a list of all weapons (by character in alphabetical order), who they belong to, how to get them and how to get the Crests and Sigils for them. After obtaining the weapon and its crest and sigil, return to the giant tree-sphere in Macalania Woods to fully power up the weapon. You will be able to visit the tree at any time, and can power up the weapons in any order.
 Celestial Weapons and How To Obtain Them
Location of Weapon: Mushroom Rock Road. Before going to Mushroom Rock Road, however, you must travel to the Calm Lands. Enter the gorge (where the Cavern of the Stolen Fayth is). Go to the right and follow the path. You will find a sword stuck in the ground, and when you observe it with the X button, you will pick it up. Now you can go to Mushroom Rock Road (enter from the Mi'ihen Highroad). Keep going north and take the second path. Use the lift to go up and there is a statue. Observe the statue to use the Rusty Sword. A mandala will appear. Press X on it and use the Mirror. Now you have the Masamune.
Mars Crest: Go to the Mi'ihen Oldroad and go all the way south to the dead end where you will find a chest with the Crest. Simple? The Crests are always simple to get.
Mars Sigil: Unlock any combination of 10 Area Creations or Species Creations bosses at the Monster Arena to the east of the Calm Lands (see Monster Arena), and the trainer will give this to you.
Difficulty Collecting: Easy and shouldn't take long to do 30 mins to 1 hour.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [First Strike] + [Counterattack]
Special Effects: Does more damage when Auron's HP is depleted, and allows Yojimbo to exceed 9,999 damage.
Location of Weapon: Thunder Plains. First you need to pray at four Cactuar Stones scattered across the Plains. They're the stones that are glowing, and press the square button to activate them. When you are done, go to the south Thunder Plains and travel along the right side until you will find a Cactuar Ghost. Follow it and it will bring you to a broken Lightning Rod Tower. Pray at this Tower by pressing the square button. Open the chest that appears for the Spirit Lance.
Difficulty Collecting: Easy except collecting the Sigil; it can be frustrating at first but once you have a few tries at it you will learn the path and will do it with no problems at all.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [Double AP] + [Evade & Counter]
Special Effects: Does more damage when Kimahri's HP is full, and allows Ixion to exceed 9,999 damage.
Location of Weapon: Baaj Temple in the pool. First you must defeat Geosgaeno. Once that is done, go to the hole in the wall opposite the door to where Anima is located. You won't be able to see the chest, but keep pressing X till it asks you to use the Celestial Mirror.
Venus Crest: Visit the Farplane a second time and on the left there is the chest that obtains the Crest.
Venus Sigil: Thunder Plains. You must complete the Lightning Dodger side quest (200 consecutive dodges). Once that is done, go to the Travel Agency and open the chest.
Difficulty Collecting: Hard, of course the Crest is easy to get, but to dodge 200 consecutive lightning bolts can be very annoying and time consuming. You need a lot of patience and the ability not to blink when doing this.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [Magic Booster] + [One MP Cost]
Special Effects: Does more damage when Lulu's MP is full, and allows Shiva to exceed 9,999 damage.
Location of Weapon: On the Airship, input the code: GODHAND, not godhand or Godhand. It must be all capitals. Then follow the path to the chest.
Mercury Crest: Go to the area just before Home and near the northwest you will find a hole. In the hole is a chest containing the Crest.
Mercury Sigil: Same area as the Crest but blocked off by a Sandstorm. To clear the sand you must complete the Village of the Cactuars side quest. Once the sandstorm clears, search the chests for the Sigil.
Difficulty Collecting: Simple, but time consuming. This one can be done quickly if you keep at it.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [Double AP] + [Gillionaire]
Special Effects: Does more damage when Rikku's HP is full, no effect on Aeons.
Location of Weapon: Calm Lands. Beat the trainer's time in the Chocobo Trainer side quest. When you go all the way to the northwest, the guy guarding the way to the Caladbolg has left his post (silly man). There will be a path going down, and a mandela at the bottom. Use the Mirror on it and receive Caladbolg.
Sun Crest: Zanarkand. I suggest getting this directly after you defeat Yunalesca. Just walk down the stairs at the end of the room (where Yunalesca was) and you will teleport to another set of stairs. Proceed back up those stairs and there will be the chest containing the Crest.
Sun Sigil: Calm Lands. You must beat the chocobo trainer with a time of 0:00 in the Chocobo Trainer side quest. It may take a couple of times. I suggest collecting 15 Balloons and don't get hit by a bird (easier said than done). You will then get the Sigil as a reward.
Difficulty Collecting: Medium. It can take a while to get the rhythm on when to collect balloons and when to dodge birds.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [Evade & Counter] + [Magic Counter]
Special Effects: Does more damage when Tidus' HP is full, no effect on Aeons.
Location of Weapon: Luca Café. Talk to the barmaid and she will ask to see the Mirror. Show her and she will give you World Champion, but only if you've been playing blitzball. If she doesn't give it to you, play a couple blitzball games and return to her.
Jupiter Crest: Luca. Go to the Aurochs' Locker Room. Search the lockers and you will find the Crest.
Jupiter Sigil: Blitzball. This can be very time consuming. First you must get all of Wakka's Overdrives (available as a Blitzball prize). Once you obtain all three, the Sigil will be a prize in the League games. You can save before competing in the 10th game, and see if the Sigil is a prize. If not, reload until it is. It's only a 50% chance of it being there, so keep trying.
Difficulty Collecting: Depends on your Blitzball skill, and remember this is very time consuming.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [Double AP] + [Evade & Counter]
Special Effects: Does more damage when Wakka's HP is full, and allows Ifrit to exceed 9,999 damage.
Location of Weapon: Calm Lands Monster Arena. The trainer will give it to you as a prize for capturing one of every monster in the calm lands and you will get the chest containing Nirvana. (See Monster Arena for more information.)
Moon Crest: Besaid. Head to the beach and get in the water. Depending when you do this, the camera angle changes. If you do this stright away when Tidus arrives at Besaid, it will be on the far right. Any time after this it will be on the far left because the camera turns around.
Moon Sigil: Remiem Temple. You must battle with Belgemine and defeat all the Aeons Including the secret aeons Anima, Yojimbo and the Magus Sisters. After you've done this, she will ask you to send her. Perform the sending and the Sigil is all yours.
Difficulty Collecting: Possibly the easiest one to do.
Abilities: [Break Damage Limit] + [Triple Overdrive] + [Double AP] + [One MP Cost]
Special Effects: Does more damage when Yuna's MP is full, and allows Valefor to exceed 9,999 damage.