Summoned mythological deities with glitzy attacks are like a Final Fantasy series tradition now. Eidolons in Final Fantasy XIII have an extra ability beyond casting Diamond Dust, they transform into vehicles.
Before you can turn Odin into a horse, you have to call him first. Final Fantasy XIII drops magic points and introduces TP or tactical points. These are used to cast special spells like Libra to scan for weakness and Quake, an earth powered attack that hits all enemies on the screen. Summoning Odin also costs TP, three TP to be exact. The amount of TP you recover after each battle depends on how well you fought.
Once Odin drops in he takes the place of your party members. Odin casts spells like Thundara, Thundaga, and Curaga automatically. See the purple meter with the number 444? That’s the summon point meter. It depletes over time, faster if Odin takes damage. While Odin is in eidolon form you want to chain attacks to increase the drive meter before the SP meter runs out.
The drive meter determines how long you can use Odin in gestalt mode. You can press square to switch into horse riding mode right after summoning him, but only nets around 16 ATB for Lightning. Chain attacks and you get more ATB, in other words more attacks with Odin as a horse. Even though Odin transforms into Sleipnir you can’t move or “drive” him. Instead you pick attacks Final Fantasy VI Blitz-lite style. Holding a direction + pressing X makes Odin do different attacks. Each attack consumes the ATB count by a set amount of points. If you don’t select an attack before the timer on the bottom left runs out you lose one ATB point. Zantetsuken, Odin’s strongest attack, uses up all of the ATB available. Probably a good idea to save that move as Odin’s final strike. (And maybe pressing select to skip it too, if you’ve seen the animation over two dozen times.)