Satan's true Demon form has not yet been shown in the series, but when he possessed Shiro, he had typical Demon features such as pointy ears, long nails and sharp canine teeth. The possessed version of Shiro also shows features similar to Rin's own Demonic form, these being the black tail with a blue flame at its end and blue flames covering the body.
The various hosts of Satan have of course possessed their own appearances. Only two people in the anime version, Yuri and Yukio, didn't get incinerated upon possession. Shiro could hold on briefly because he was created as a vessel for Azazel.
When first introduced, Satan's mannerisms and laughter bordered on insanity and he enjoyed it when Rin screamed for help while being sucked into the Gehenna Gate. This reveals that his cruel and reckless personality doesn't stop even before his own family. It is further emphasized when Satan said that he created Rin out of mere boredom and after that realized that Rin is exactly what he needed to conquer Assiah as Rin is a human born in Assiah but as Satan's offspring bears Satan's blood.
In the anime he is shown to have been in love with Yuri and seeking to realize her dream of humans and Demons coexisting he attempted to merge Assiah and Gehenna. Ironically enough, he shows no remorse for possessing Yukio and trying to kill Rin even though Yuri loved both of her sons more than anything.
Satan was first brought into Assiah 16 years ago. His ego sprouted into the strongest clone the research facility made, that clone was meant for Lucifer but unbeknowst to the other Demon Kings, they had no idea who it was that possessed the body meant for Lucifer. By the time they realised, it was too late. The Blue Night happened that same day, where Yuri Egin was impregnated by Satan and the mass slaughtering of exorcists around the world took place. 
True Cross Academy arc
Taking advantage of Shiro Fujimoto's temporary weakness after his argument with Rin Okumura, in which the latter told the former that he shouldn't pretend to be his father, Satan possesses the priest. Grinning, he muses on how he has waited for the day when he would finally see his son, before breaking into hysterical laughter and denying the entire thing. He then tells Rin to be happy, as he has come all this way to see him in person. When the boy realizes that he is a Demon, Satan introduces himself, but remarks that he doesn't have time to continue their "touching reunion", as Fujimoto's body won't last much longer under the current circumstances. He then rips off several of the priest's fingers to illustrate his point, using the blood from the severed digits to summon a Gehenna Gate. He tells Rin that he is the only Demon that is able to create this apparatus, and that he will use it in order to drag Rin back to Gehenna with him.
Taking note of Kurikara, Satan picks up the sword, informing Rin that he will need to destroy it first before they can make their trip. When the teenager reacts violently to Satan's approaches by summoning his flames, the Demon king merely laughs off the attempt, grabbing him by the collar and dragging him over to the Gate. As he does so, he explains how even with his supreme power, he is unable to remain indefinitely in Assiah, as everything he possesses ultimately is destroyed. Stating that it was nothing but a whim that prompted him to create Rin, he goes on to say how much of a masterstroke that action actually was, as he now has exactly what he needs to make Assiah his. He then tosses Rin into the Gate, laughing uproariously as he declares that the boy will be reborn, welcoming him to Gehenna. Before Satan's plan could be completely successful, however, he is stopped by the remaining consciousness of Fujimoto, and curses the priest as he temporarily seizes control of his body, grasping his Exorcist pin and thrusting it into himself. As Fujimoto claims Rin as his son, and that he is going to take him back, Satan gleefully exclaims how wonderful it is that Fujimoto plans to commit suicide, as no matter what he does, nothing can escape the Gehenna Gate. After this, Satan is sent back to Gehenna as Fujimoto dies, leaving Rin to escape the Gate by himself.
Powers & Abilities
As the God of Gehenna, Satan is the strongest Demon in existence. His powers are not yet revealed except that he can summon the Gehenna Gate. Like any other Demon, Satan has the power to possess a human body but only for a moment as everything he touches in the world of Assiah is doomed to destruction, with even the strongest bodies only lasting five minutes at most
As the God of all Demons, Satan possess immense strength, giving him the ability to carelessly drag someone like Rin with ease and rip off his own fingers with no difficulty.
Satan has the ability to summon the Gehenna Gate; in the anime, it is implied that he's the only being in Gehenna powerful enough to summon and open the complete and full form of the Gehenna Gate, although other Demons are able to by conjuring a lesser or an incomplete version of the gate. This Demonic door needs blood in order to link Gehenna and Assiah. It's size, appearance, and even abilities depend on the amount of blood offered as a sacrifice.
Being his blood, Rin was sacrificed by the True Cross Order to use his blood to create a large Gehenna Gate which requires a large amount of Demon blood. Combining Yukio's blood with Rin's, the power dramatically increases, creating multiple Gehenna Gates to create a large portal to Assiah.
After his defeat from Rin and Yukio, Satan was rejected from his vessel, but possessed and took control of Gehenna Gate, giving more abilities to send gate Demons, shoot beams from its eyes, and rise into the sky so it can attack the city. This Gate is regenerative and able to withstand Rin's flames. Its weakness however is its heart which Yukio and Rin pierce through to destroy Gehenna Gate completely.
Like Rin, Satan is highly durable; maybe even more than Rin, due to his being Rin's father and a demon. It was shown that he could stand a fight against his son's friends without feeling exhausted. The same thing happened when he faced Shura's power as well. However, unlike Rin who cannot use his flame in a long period of time with ease and has to practise for that, Satan does not seem to have any difficulty in this.
Just as his son, Satan can control and manipulate blue flames at will. He is able to ignite his victims in blue flame by spontaneous combustion (activated by eye contact). He also has the ability to shoot multiple fireballs from his palm during his battle with Shura and can burn out any barrier that was created by the Exwires. When fighting Rin, he ignites him with his flames but Rin, however, appears immune to it. This was only shown in the anime.
Satan has the special ability to generate powerful wind from his own voice. It was shown in the anime when he told Rin to 'shut up'; a great wave of hurricane suddenly emerged from his deep voice. He did the same when he knocked out all Rin's friends as they were trying to fight him.
Satan is a very fast opponent but the limits of this ability has not been revealed yet. However, it has been seen that when he was trying to attack Rin, he initially laughed at him before using a high speed rush to get to him; a proof that he may have this kind of ability.
- In the Abrahamic Religions, Satan is the name given to a fallen angel who fell out of God's favor after waging a war in Heaven. He is also the personification of all things evil, as he leads humanity astray with deception and temptation.
- (To Rin Okumura) "I'm a perfect entity with ultimate power, but there's one thing I lack. Do you know what it is? It's a substance in Assiah strong enough to contain me! Everything in this world that I touched is doomed for destruction!"
- (To Shiro) "Curses what do you think you're doing? ...So you mean to sacrifice your life for him? You astound me, you wretched priest. However, it's already too late. Gehenna Gate has taken a hold of you, and it will never let you go."
- (To Rin Okumura) "Oh, it was out of mere boredom that I created you... but I have since realised, it was a masterstroke! You exist in the world of Assiah... and yet you bear the blood... of the god of Gehenna...! You are exactly what I need to make Assiah mine!!"