[Give her a second because how the hell does she explain eidolons...]

They're like...when people get chosen by the fal'cie, well before all the fal'cie died, they were given a focus and magic powers. If it seemed like they were faltering in their focus, their Eidolon would appear and try to kill them. Of course.

The fal'cie were basically jerks.

If they won, they got control over it. Lightnin's eidolon was Odin.