Affably Evil/Faux Affably Evil: Zigzags the two. To most people, he's Faux Affably Evil, often utilizing Exact Words to make people think he's about to give them what they want before making their torments even more painful. To other supernatural beings and the handful of humans he respects, he's Affably Evil; he'll still torture them if they become annoying, but when he's not doing so, he's fairly friendly and can sometimes even act as a caring presence.
Even Evil Has Loved Ones: He gets a one-shot love interest named Hope in one Christmas special. He also cares more about Scumspawn and the Professor than he lets on.
Evil Redhead: Possibly. Since it's revealed in Series six that all redheads are of angel descent and vice versa, it stands to reason that all angels are also redheaded.
Green-Eyed Monster: The Professor often notes (to the point of it being a catchphrase) that Satan torments humans because he wants to bring everyone down to his level- he hates himself and his situation and wants everyone to be as miserable as he is. For extra points, Satan is actually described as having green eyes.
Hypocrite: One of the main reasons for his hatred of humans is their refusal to own up to their mistakes, yet he continually objects to any attempt to blame him for his own rebellion against God and subsequent banishment from Heaven.
Paper-Thin Disguise: He may be able to change his shape, but humorously, doesn't do much (or often anything) to change his voice. At one point he complains that God saddled him with inflexible vocal cords, so it's likely he actually can't.
Wouldn't Hurt a Child: Satan has strict standards of not possessing anyone who isn't old enough to make moral choices, and is thus in one episode is shown to be enraged at fundamentalists who believe a child is possessed by him, and want to beat Satan out of the child.
Flanderization: To an extent. In the first series, he's more of a malevolent cad, but later episodes present him as a rage-filled psychopath. Weirdly, this happens at the same time as efforts are made to reform him.
Not-So-Harmless Villain: He's more of a punching bag than a villain by the time of the series. However, in life, he murdered, raped, and tortured god knows how many innocent people, and was able to get away with it completely until Satan found out about it.
Determinator: As Satan notes 'arguing with her is like... well it's like cornering a badger'.
Suspiciously Similar Substitute: introduced after the Professor leaves and plays a similar role. Played with in that there are notable differences between the two characters; where the Professor's idealism comes from trying to find the bright side of every situation, Edith's comes more from romanticizing certain people, ideas, and events while seeing no good in others. Also, while The Professor came to be fairly well-liked even by the villains, Edith is tolerated at best by most of the other characters.
Shrouded in Myth: She's absolutely nothing like you'd expect from her work, or even from historical accounts.
Never My Fault: He maintains that the Holocaust was merely overreacting to being filmed in the shower, and he blames his failed invasion of Russia on Jesse Owens.
Ultimate Evil: He's the most frequently mentioned out of all the historical figures, but he never speaks, is never spoken to, and other than one scene where he is explicitly in disguise, is never even spoken about in such a way that one can infer his presence.