Since I believe I recently acquired the final Eidolon, I have one for each main character, I will talk about the eidolon system in FF XIII. So to acquire an eidolon you must earn its trust by fighting it. The first battle against the Shiva twins was simple, one twin kept you healed while you fought them by doing things to make their gestalt gauge go up by doing specific things for each character: Snow increased the gauge faster by using the Sentinel paradigm. A twist in this is your character is inflicted with doom so you have like 1500 seconds to do this, and the Shiva battle is only in which the eidolon heals you.
I am a little upset how they seemed to take some of the classic eidolons and make them mechanical, but I do admit they look cool. The ones that are new like Alexander and Hecatoncheir are pretty cool. I really like how Alexander is a floating fortress in gestalt mode. That brings me to the battle system, when the eidolon is first summoned you battle alongside it while its "health" steadily goes down, faster if it, and building up the gestalt gauge, when you think it is high enough and before its health goes all the way down press square and enter gestalt mode in which you ride the eidolon. In this mode you choose various modes that use up the gestalt and when empty the eidolon leaves and you and your party is fully healed and ready to finish the battle if it is still going!