Eidolon (Final Fantasy IX)

Summons in Final Fantasy IX are referred to by the term Eidolon. They can only be summoned by members of the tribe that once lived in Madain Sari. After the destruction of Madain Sari, only Princess Garnet and Eiko Carol remain as members of that tribe.

The spell power of an Eidolon is based on math, item stock and length of animation, i.e. short or full length. The full length animation can be regarded as a "powered up" version of the short animated version. In total, there are 12 Eidolons. Of these 12, 8 can be summoned by Garnet and the remaining 4 by Eiko.

Garnet's Summons
The attacks and spell power for Garnet's summons are pretty straight-forward. They are based on math, item stock and length of animation, the short version being weaker in spell power than the full length animation (FLA). When summoning in Trance mode (TM) no MP is used for the summon, but the length of the animation will always be short and item stock "bonuses" will be disregarded. However, the summon will reappear in intervals to repeatedly damage the enemy. This continues until a new Eidolon is summoned.

Shiva
Shiva can be obtained through the use of an Opal. Its attack is called Diamond Dust and deals ice-elemental damage to opponents.
 * Spell power (FLA) equals 36+# of Opals in stock
 * Spell power (TM) equals 34

Ifrit
Ifrit can be obtained through the use of a Topaz. Its attack is called Flames of Hell and deals fire-elemental damage to opponents.
 * Spell power (FLA) equals 42+# of Topaz in stock
 * Spell power (TM) equals 37

Ramuh
Ramuh can be obtained through the use of a Peridot. Its attack is called Judgement Bolt and deals lightning-elemental damage to opponents.
 * Spell power (FLA) equals 32+# of Peridot in stock
 * Spell power (TM) equals 31

Atomos
Atomos can be obtained through the use of an Amethyst. Its attack is called G-Force 199 and damages an enemy equal to a certain percentage of their Max HP (total damage is capped at 9,999).
 * HP reduction (FLA) equals (30+# of Amethyst in stock)%
 * HP reduction (TM) equals 16%

Odin
Odin can be obtained through the use of a Dark Matter. Its attack is called Zantetsuken and deals Instant Death. If Garnet's "Odin's Sword" ability is equipped, Odin will deal non-elemental damage to all surviving enemies as well. When summoned in TM, he does non-elemental damage, disregarded if Odin's Sword is equipped or not.
 * Chance of Instant Death (FLA) equals (# of Ore in stock/2)%
 * Spell power of "Odin's sword"-attack (FLA) equals 142-#of Ore in stock
 * Spell power (TM) equals 49

Leviathan
Leviathan can be obtained through the use of an Aquamarine. Its attack is called Tsunami and deals water-elemental damage to opponents.
 * Spell power (FLA) equals 59+# of Aquamarine in stock
 * Spell power (TM) equals 53

Bahamut
Bahamut can be obtained through the use of a Garnet. Its attack is called Mega Flare and deals non-elemental damage to opponents.
 * Spell power (FLA) equals 88+# of Garnet in stock
 * Spell power (TM) equals 77

Ark
Ark can be obtained through the use of a Pumice. Its attack is called Eternal Darkness and deals shadow-elemental damage to opponents.
 * Spell power (FLA) equals 106+# of Lapis Lazuli in stock
 * Spell power (TM) equals 87

Eiko's Summons
Eiko's summons are a little more complex, as the attacks/effects of her summons all change according to different variables, such as equipped add-on.

Fenrir
Fenrir can be obtained through the use of a Sapphire. Its attack is add-on dependent. If normally summoned, its attack is called Terrestrial Rage and does earth-elemental damage. If Maiden Prayer is equipped, it attacks with Millennial Decay, which does wind-elemental damage.
 * Spell power for Terrestrial Rage equals 42+#of Sapphires in stock
 * Spell power for Millennial Decay equals 44+#of Sapphires in stock

Phoenix
Phoenix can be obtained through the use of a Phoenix Pinion. Its attack is animation length and chance dependent. It's attack is called Rebirth Flame and does light to moderate damage to enemies while reviving KO'd allies. If entire party is KO'd, there's a slight chance of auto-summoning. This revives the party, but does no damage to enemies.
 * Amount of Max HP regained on revive (FLA) equals 30+revived character's spirit stat
 * Chance of Auto-summon equals #of Phoenix Pinions/256

Carbuncle
Carbuncle can be obtained through the use of a Ruby. Its attack is add-on and animation length dependent. Unlike other summons, it does no damage to the enemy, it only casts helpful status effects on the party. The difference between short animation (SA) and full length animation (FLA) is that if the FLA is played, Protect will be casted in addition to the normal status effect.
 * Normal summon "attack" is called Ruby Light and casts Reflect on the party
 * If Diamond is equipped, "attack" is called Diamond Light and casts Vanish on the party
 * If Emerald is equipped, "attack" is called Emerald Light and casts Haste on the party
 * If Moon Stone is equipped, "attack" is called Pearl Light and casts Shell on the party

Madeen
Madeen can be obtained through the use of a Ribbon. Its attack is called Terra Homing and does holy-elemental damage.
 * Spell power equals 71+Eiko's level

Extraction
The Eidolon Extraction is a process in which the Eidolons are forcefully extracted from a subject (Princess Garnet, in the game). Zorn & Thorn can be seen attempting the extracting ceremony twice in the game: firstly they call forth Eidolons from Dagger, so that Queen Brahne can make use of her daughter's eidolons to take over the nations of Cleyra and Lindblum. Later in the game they attempt to steal Eiko Eidolons for Kuja but they are unsuccessful.

Garnet is captured and imprisoned by Zorn and Thorn when she returns to Alexandria to reason with her mother about her recent attack on Burmecia. Queen Brahne lies to Garnet and Kuja then puts her to sleep so the extraction can begin. After Garnet loses her Eidolons she is unable to use them in battle until the party encounter Ramuh and she learns Judgement Bolt.

Later, when the party is at Kuja's Desert Palace, Eiko is captured. Kuja and the buffoon twins escape on the Hilda Garde to Esto Gaza. The Black Mage army, Kuja and Zorn and Thorn then enter Mount Gulug where they intend to extract her Eidolons as Kuja says he needs an Eidolon more powerful than Alexander.

However, they fail to extract Eiko's Eidolons. At first Zorn and Thorn state that the ceremony can only work when the subject is sixteen years old or they risk endangering her life. Mog then appears and attacks the twins, preventing the ceremony from going any further, Mog goes into trance and tells Eiko to summon Terra Homing. It is then revealed that Mog is in fact the Eidolon Madeen who became Mog to protect Eiko.