Setting/World

Assiah.
Assiah is the material world in which humans live. Demons can pass into this world by possessing anything that exists within it. It is the ultimate goal of Satan, the ruler of Gehanna, to make Assiah his own. Specially trained humans in the world of Assiah can exorcise demons back to Gehenna through varioous methods. Any kind of contact between the two worlds is normally impossible to save Demons who live in Assiah by possessing objects.

Mashō
Mashō or Temptaint, refers to the wounds or illnesses that are inflicted by a Demon. once a person sustains a Mashō, they become able to see demons for the rest of their life. As such, it is a rite that all propective Exorcists must go throughm inbluding Yukio Okumura. Evidently, Rin did not have to undergo a mashō because of his demon heritage.

In the anime "mashō" was renamed "Spirit wound".

Gehenna.
Gehenna is the world in which Demons live. Satan is the god of Demons and the ruler of Gehenna. Currently, not much is known about Gehenna, as it has not yet been seen. It has been stated that there is no material present in Gehenna. It is said that the only direct way to the Demon world is through the Gehenna Gate. Which only Satan can open.