Fang: Yeah, ‘help.’ That’s what Eidolons do, they help us. Eidolons are our salvation. If we can’t decide what to do next, they come put us out of our misery!
Each character’s eidolons represent their problems. In this sense, the pantheon of these mystical entities exist in their minds, with their hopes and feelings:
Snow’s eidolon Shiva represents his relationship with Lightning and Serah. Nix, the one who will attack Snow, represents Lightning. Stiria, who represents Serah, will heal him. You can fight back, but neither of the sisters want that. Lightning needs someone to take out her frustration on and Serah wants Snow to not fight back. Snow wins by taking it instead of dishing it out.
Since Snow fights his Eidolons alone, Stiria cannot be targeted during the fight, but instead heals Snow as he has no other support. Shiva Sisters “yield to those who defend against attacks”, so the simplest method of victory is to Paradigm Shift to Sentinel, and repeatedly use Steelguard.
Odin represents Lightning, who can’t protect and only destroys. This is how Lightning views herself, and blames herself for not being able to help her sister. Lightning can defeat Odin by protecting Hope (who’s also caught in the battle). Like Lightning:
Odin is an aggressive Eidolon, initially targeting Hope with physical attacks and Thundara. Lightning can satisfy Odin by building chain bonuses and healing, and Lightning will need to do both as Odin’s strikes will necessitate healing greater than Hope alone can supply. Odin will often guard with Ullr’s Shield, allowing for either a break to heal or an uninterrupted attacking phase.
Sazh’s eidolon Brynhildr represents his rage. Sazh becomes enrage over finding out Vanille’s involvement with his son becoming l’Cie drives him to fury. The rage is enough to kill both Vanille and him. He can win by helping Vanille.
Brynhildr is fought by Sazh and Vanille in Nautilus Park The Fiendlord’s Keep area, where Brynhildr manifests as a result of Sazh and Vanille’s confrontation escalating. Brynhildr attacks with fire attacks that hit both Sazh and Vanille. The key to winning the battle is employing Sazh as a Synergist and only switching to offensive roles when all buffs he has available are in place.
Hope can defeat his eidolon, Alexander, by fighting a gigantic robot to prove his worth to the team.
As Hope loses his drive to continue and asks the others to go on without him, he unwittingly summons Alexander, who turns against him, but Lightning and Fang rush to his aid. Hope can win against Alexander by acting as a Synergist and healing the party when they are on low HP, and acting as a Ravager during times no support is needed.
Remember when Fang pushed Vanille away to face her problems alone? Fang thought she could find a solution to hers and Vanille’s Focus. She can’t defeat Bahamut alone. She needs help from her friends.
Fang can appease Bahamut by fighting as a Saboteur and switching to Sentinel to guard against Bahamut’s strongest attacks.
Vanille’s eidolon Hecatoncheir and its arms represents her many lies that she used to grapple others into her problems. Only when she can face her problems, instead of running away from them, can she defeat Hecatoncheir.
The battle commences after entering the area when Vanille and Fang talk about their role in the War of Transgression and Fang discovers Vanille has been lying to her ever since they woke up from crystal stasis. Vanille’s emotional turmoil calls forth her Eidolon, that attacks the pair. Fang should remain as a Sentinel for most of the battle leaving Vanille to debuff as Saboteur and heal and use Ravager when neither are needed. When Hecatoncheir is charging, the party can use offensive paradigms, but Fang should switch to Sentinel when he starts attacking.
In a sense, they’re all more like guardian angels and teachers. They impart wisdom with how to deal with their problems. I wonder how many more eidolons there are? Do the eidolons represent problems that l’Cies have, or do they come into existence once the l’Cies have problems. Thematically, I include the party members that best complement their problems (even though we can’t always given a choice).
These mystical entities reveal themselves before only to a select few l’Cie. It is said that they are saviors, come to rescue hopeless l’Cie who find themselves bound to a Focus against their will. If this is true, they offer a brand of salvation few indeed would seek willingly: without exception, Eidolons attack the l’Cie whose presences they grace. There have been no l’Cie in Cocoon for centuries, and as a result, no way of determining the truth behind tales of these beings. To the citizens of Cocoon, they remain the stuff of bedtime stories.
Source, TVTropes: To win an Eidolon battle, you need to fill the ATB gauge before the Doom counter hits zero, and certain actions will make it fill more quickly than other actions. Basically, the character has to show his or her Eidolon their worth.