FFX Aeons

Valefor

Acquirance: You will receive Valefor after the Besaid Temple.
Overdrive: His original overdrive is Energy Ray, but there is another overdrive he can learn named Energy Blast. To obtain this, return to Besaid after getting Yuna into your party. Speak to the girl with a dog standing next to the temple, and you'll receive an item that gives Valefor the second overdrive. If the little girl isn't near the temple, then look for her inside the item shop.
Battle Strategy: Valefor doesn't really have any bad battle situation, unless the enemy is a hard hitter. He's especially useful, however, against grounded enemies.
Unique Attack: Sonic Wings. This attack only hits one enemy, but not only attacks him, it also delays that enemy's turn. Because of this delay effect and the short recovery time from using this attack, Valefor can often hit enemies repeatedly with Sonic Wings.
Break Damage: To break the 9999 hit limit, activate Yuna's legendary weapon, Nirvana.

Ifrit

Acquirance: During the Kilika trials, you will get Ifrit.
Overdrive: Hellfire. Ifrit encases its foes in a swirling ball of fiery vengeance and detonates it for massive damage to all enemies.
Battle Strategy: Ifrit has very high strength, but low Defense and Magic Defense. But you won't need to worry about dying too much, considering all you have to do is cast a fire spell on him to cure him. He's best used against slower enemies, so that he can get those big hits in before his low defense gets the best of him.
Unique Attack: Meteor Strike. Meteor Strike is non-elemental that is directed at one enemy, but has a longer recovery time than Ifrit's regular attack. This is a useful long-range attack. It also penetrates Protect despite being a physical attack.
Break Damage: Once you get Wakka's legendary weapon, World Champion, Ifrit can do more than 9999 HP damage.

Ixion

Acquirance: Ixion becomes available in the Djose Temple.
Overdrive: Thor's Hammer. Ixion envelops all enemies in a storming vortex, and then disintegrates it. Lightning-based elemental damage is inflicted against all enemies.
Battle Strategy: Due to Ixion's high defenses, it's best to summon this aeon when the party needs protection from a big attack.
Unique Attack: Aerospark. This aeon's special attack attacks a single enemy with non-elemental damage, while also destroying magical effects like Protect, Shell, Reflect, Haste, Regen, and Nul spells that were on the selected enemy.
Break Damage: Ixion can inflict more than 9999 HP damage once you've activated Kimahri's legendary weapon, the Spirit Lance.

Shiva

Acquirance: You acquire Shiva at the Macalania Temple.
Overdrive: Diamond Dust. Shiva freezes all enemies in layers of ice, and then destroys everything with a leisurely snap of her fingers.
Battle Strategy: Well, I suppose it's best to use this ice goddess when you're at a battle for speed. She's got amazing agility, so those slow attackers won't last long at all.
Unique Attack: Heavenly Strike. Inflicting non-elemental damage against one enemy, this attack slightly delays the enemy's turn, while also inflicting the status ailment threaten.
Break Damage: After acquiring Lulu's Onion Knight, her legendary weapon, all of Shiva's attacks can do over 9999 HP.

Bahamut

Acquirance: This awesome aeon comes into your power in the Palace of St. Bevelle.
Overdrive: Mega Flare. Bahamut anchors itself and fires a destructive breath attack that annihilates anything in its path.
Battle Strategy: This is the aeon you will use most if you don't go for the secret ones. He's got amazing strength, defense, and magic attributes, though his agility is rather low. He's best used against those giant hard hitters that are just as slow as Bahamut.
Unique Attack: Impulse. Bahamut fires away, hitting all enemies.
Break Damage: Bahamut comes equipped with the Break Damage ability, meaning he can hit for over 9999 HP right at the git-go without the aid of a legendary weapon.

Yojimbo

Acquirance: This secret aeon can be obtained at the Cavern of the Stolen Fayth, located beneath the bridge at the northeast end of the Calm Lands. In the Calm Lands, circle past the right side of the bridge leading to Mt. Gagazet, and descend into the gorge below. To the left of the Save Sphere is the Cavern of the Stolen Fayth. From the entrance, go right. Follow that trail, picking up the Megalixer on your way, until you encounter a T-intersection. Go right to get a Level 2 Key Sphere, then turn around and go left until you encounter another intersection. Go north at the intersection to get a Fortune Sphere, and a save sphere as well. Then go left to find an the 25th Al Bhed Primer, as well as a Mega-Potion. Return to the Save Sphere, and head north until you encounter the summoner. The summoner will call Yojimbo. The best strategy is to hit him as often as possible with spells such as Firaga. Afterward, you can use the teleport pad there to travel to hidden chambers containg a Flexible Arm, an MP Sphere, and two X-Potions. Next, travel to the Chamber of the Fayth and speak to Yojimbo to negotiate a fee for his services. He will ask you a question about why you wish to hire him. He will offer his lowest initial asking price if you choose the option "To defeat the most powerful of enemies." For your first bid, offer half of his asking price plus 1 gil. So, if he asks 250,000 gil, offer 125,001 gil. After his next offer, raise your bid by 1 gil. Which means, to the best of my knowledge, that if you offered 125,001 gil first, now offer 125,002 gil. By the time he proposes 205,000 gil, he will usually accept offers below 200,000 gil, though not much lower (I'd be safe and just go with 200,000). If at any point you offer him at least triple his asking price, he'll give you 2 Teleport Spheres in addition to his services.
Overdrive:None. Yojimbo doesn't have one, so don't waste Yuna's Grand Summon on him.
Battle Strategy: Uhh... you can't control Yojimbo, so there is none.
Unique Attacks: Daigoro, Kozuka, Wakizashi, and Zanmato. The attack that he does depends on how much money you give him. Personally, I find this aeon quite useless, except for the fact that you have to have him to get the Magus Sisters. The money it costs to pay this guy is odd, because I paid him 200,000 and all he did was his dagger-throwing ability (Daigoro, I think it is).
Break Damage: Once you get Auron's legendary weapon, the Masamune, Yojimbo can do more than 9999 HP damage.

Anima

Acquirance: Anima, yet another secret aeon, can be found at the Baaj Temple. Once you get the airship, put in these coordinates: 13, 60 (for those of you that don't know X from Y, X is 13 and Y is 60). Once there, run up the path and dive down into the water. Then prepare to defeat Geosgaeno, and swim towards the underwater temple entrance. The best way to defeat this monster is to equip armor protecting the party from petrification and death. Cast Hastega and Protect on the party, and use everyone's overdrives. After killing him off, swim north into the temple entrance. Enter the Chamber of the Fayth, where you'll find four Mega Phoenix itmes and a Megalixir. Approach the 6 statues on either side. There is one hidden item in each of the 6 Cloisters of Trials. You should have 5 of these items at this point, and five of the statues in the room will activate as a result. Now you just need to get the last item from the Zanarkand Dome. Fly back to Zanarkand and enter the dome. A new puzzle solution will be displayed on-screen. Step on the three white squares in the first room and the four in the larger chamber. This opens a panel in the first room revealing a Destruction Sphere. Place the sphere into the slot to the right of the monitor in the larger chamber, and the screen will explode. Collect the Magistral Rod, and return to Baaj Temple. Return to the room with the statues, where the statues will allow you entrance to the Chamber of the Fayth. And that's it; Yuna retrieves the aeon. If you need help finding the other items in the other temples, check out the "Temple Help" section
Overdrive: Oblivion. In the most horrifying (and most awesome) of all the aeons' Overdrives, Anima breaks from his chains to inflict multiple non-elemental attacks on all enemies.
Battle Strategy: Anima is amazing. Plain and simple. Use him especially against enemies that are not immune to death (Pain will kill them instantly).
Unique Attack: Pain. In this attack, Anima attacks one enemy, with a high chance of causing instant death. Unfortunately, this attack has a high recovery time.
Break Damage: Anima can cause more than 9999 damage from the get-go.

Magus Sisters

Acquirance: This is the hardest aeon to get in the whole game, and requires lots of work. So, I'll just start at the beginning. First, capture all of the required types of fiends on Mt. Gagazet (I think you only have to catch one of each type of fiend) and return to the monster arena owner in the Calm Lands. As a reward, he gives you the Blossom Crown. You must also possess all of the other aeons, including Yojimbo and Anima. Next, go to Remiem Temple in the Calm Lands. Once there, defeat Belgemime's Valefor, Ifrit, Shiva, Ixion, and Bahamut, and she will give you the Flower Scepter. After that, use the Blossom Crown and Flower Scepter on the door in the back to unseal it and acquire the Magus Sisters.
Overdrive: Delta Attack. When the Magus Sisters are all in Overdrive mode at the same time, a new command, "Combine Powers!" appears. The sisters will join forces and perform one of the most devastating Overdrives in the entire game, encasing their enemies in a pyramid and shooting multiple non-elemental energy rays at the victim.
Battle Strategy: Boy, do we have some unique ones here. Even though you summon these sisters as one aeon, they're more like 3. To use these aeons, you give them verbal requests. They are:
Command Effect
Do what you will. Sister makes the choice. Other commands open.
Fight! Sister performs regular attack.
Go! Go! Sister performs unique attack or casts magic.
One More Time! Sister repeats the last action performed.
Defense!/Are you okay? Sisters cast protective or support magic.
Help Each Other! Sisters use healing magic on each other.
Dismiss Sisters leave the battlefield.

It's all up to you what they do, but just because you select one command doesn't mean that's the one they do. It depends on their level of morale. (Beware though, Cindy will usually just "Take a break")
Unique Attacks: Camisade, Razzia, Passado. To get the Sisters to use these attacks, choose "Do what you will" or "Go! Go!", even though neither of those will assure that that's what they do. Cindy's Camisade attacks one enemy, Sandy's Razzia causes MASSIVE damage to one enemy, and Mindy's Passado attacks 15 times consecutively against one enemy.
Break Damage: These aeons can do more than 9999 from the get-go. (Sandy's normal attack does around 83,000 to normal enemies, oi)