Rahxephon explained
The Mulian world MU and Earth are parallel worlds overlapping that will eventually destroy each other. This is a natural phenomenon. Renegade Mulian scientist Barbem made the Xephon System which consists of two robots, Rahxephon and Belxephon. The system can rewrite the clashing worlds into one, saving it from destruction, and activates through a ritual in which the two robots fight, and the winner's Ollin (pilot) decides what the resulting world will be like subconsciously. This is the tuning. Ayato is Rahxephon's Ollin, while Kuon (his biological mother) is Belxephon's. The stuff about them and Itsuki should be clear enough from the show. Yoloteotl is the state of the Ollin becoming worthy of the Xephon and enabling it to participate in the ritual. The Ixtli are manifestations of the Xephons' will (never seen in the show, but Belxephon/Kuon's Ixtli takes on Ayato's form). The two factions, Earth and MU, think that the tuning results in one world surviving and...