Eidolons

Eidolons are the Final Fantasy IX name for the summon monsters (they've also been called Guardian Forces, Aeons, and Avatars, among other things). Eiko has a total of four that she can summon in the game, all of which are pretty useful. There's a list below.

Fenrir
Effect: Earth elemental attack on all enemies--a giant fist emerges from the earth and hits them into the air.
Equip: Sapphire
Where to get a Sapphire: Eiko is initially equipped with one. You can find one in the rubble near the entrance of the city after Alexandria is leveled--check to the left. They can also be synthesized.
About: In Norse mythology, Fenrir is the demon offspring of Loki, taking the form of a wolf. Fenrir has appeared in two other Final Fantasy games, to my knowledge, but is an Earth elemental in neither. In Final Fantasy VI, Fenrir's power gives the party the multiple images effects, making them harder to hit. In Final Fantasy Online, Fenrir is the Darkness elemental summon.

Phoenix
Effect: Fire elemental attack on all enemies, and all downed allies are revived with full HP.
Equip: Phoenix Pinion
Where to get a Phoenix Pinion: You start the game with one in your inventory. They can also be stolen from SoulCage, the Earth Guardian, and the flying Veteran enemies (among others). They're also in abundance when playing Chocobo Hot and Cold.
About: A Phoenix is a mythological immortal bird, who dies in a burst of flame and is then reborn from the ashes. Phoenix has also appeared in Final Fantasies VII and VIII. The usage in VII is quite similar; in VIII, holding a Phoenix Pinion might summon the bird after your party is defeated.

Carbuncle
Effect: Gives all party members Reflect status.
Equip: Ruby
Where to get a Ruby: Eiko receives one when she frees Carbuncle at the Iifa tree. Others can be found in the field (often in rubble) or synthed.
About: Biblically, a carbuncle often refers to a gem, often a garnet--Final Fantasy has obviously chosen to use ruby. A Carbuncle is also a term for a mythical creature often rumored to be sighted by Spanish conquistadors. It is described as having a gem on its forehead, and as either a bird or a mammal. Obviously, in this case it's a mammal. Carbuncle appears in Final Fantasies VI, Tactics, VIII, and Online. In each of these games but Online, its effect is the same. In Final Fantasy online, it is the light elemental summon, and also the only summon that your character doesn't have to earn.

Madeen
Effect: Holy elemental attack on all enemies.
Equip: Ribbon
Where to get a Ribbon: Mog will give it to you. It might also be able to be synthed. (I don't remember. Remind me to check.)
About: Madeen looks a lot similar to the Maduin summon from Final Fantasy VI, which is said to be based on the hero of Irish folklore, Mael Duin. In Final Fantasy VI, Maduin does an all-enemy non-elemental attack, instead. What's also interesting is that in FF6, Maduin is the father of Terra/Tina, and here in FF9, Madeen is the closest friend to Eiko, so possibly there is a correlation there.