Abraham Lincoln debuts in the Animated Short, where he encourages Finn to believe in himself after warping his brain in the past and to Mars. He is also seen in "Business Time," where he is a character included in a game on BMO.
He is referenced in Issue 3 of the Adventure Time comic. When Finn is telling everyone how to beat The Lich, he says that just like Abraham Lincoln said in the past and on Mars, you need to believe in yourself.
The first time he ever appeared in the official series is in one of the many pictures in Hunson Abadeer's house. He was a major character in the episode "Sons of Mars," and is shown to be the King of Mars. He lost his immortality (and seemingly his life) in the episode in order to save Jake after he was accidentally sent to the 37th Dead World, reverting back into the statue from the Lincoln Memorial.
Abraham Lincoln looks pretty much exactly like the real-life Abraham Lincoln. He wears a full tan-brown suit with a dark red bowtie. Two major differences from his real-life appearance are that, in the show, he floats like Marceline, and he has a yellow light surrounding his head that is probably meant to signify spiritual importance. He also doesn't wear his stovepipe hat. Unlike most characters, his hands have five fingers. He sits atop a throne that is ingraved with the symbol ♂ , the astrological symbol of mars, which has come to also represent "male" in contemporary popular culture. This symbols can also be found in the arena in which Abe Lincoln sits.
It doesn't matter. What does matter is you need to believe in yourself!
- Pendleton Ward stated on his Formspring that Abraham Lincoln might appear in Season 2. Beyond a brief mention, this did not come to pass.
- As stated by Rebecca Sugar, Abraham Lincoln is the equivalent of Jesus in Ooo. However, according to Adam Muto, he is more of a "recognizable figure," while adding that it would be "a stretch to call him a religious figure." 
- He has a very detailed face, unlike most Adventure Time characters.
- Abraham Lincoln already gave Death a penny, that's why Death said "not this time" meaning he saved someone else from the 37th Dead World before Jake.
Wikipedia has an article on Abe Lincoln.