The World of Ao no Exorcist is the main setting of the series. While the series is set in modern times, the story places our world next to a parallel world: Assiah, the human world, is connected to Gehenna, the Demon world.
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 Gehenna, to make Assiah his own. Specially-trained humans in the world of Assiah can exorcise Demons back to Gehenna through various methods. Any kind of contact between the two worlds is normally impossible, except for Demons who live in Assiah by possessing objects.
Mashō (魔障Mashou), or Temptaint, refers to 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 prospective Exorcists must go through, if they had not done so incidentally prior to becoming an Exorcist. Evidently, Rin did not have to undergo mashō because of his Demon heritage.
In the anime "Mashō" was renamed "Spirit Wound". In the dub of the anime, it's referred to as a "Temptaint"
Gehenna (虚無界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 hasn't been seen yet. 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.