To Quentin, Margot refers to herself as a "paper girl" and says that the place for her is in a "paper town." Paper towns are towns that do not exist in the real world as actual towns but instead act as safeguards against counterfeit maps. Agloe, New York, is one of these paper towns, which exists as a town only on a certain brand of map for copyright purposes. Margot never felt as if the town in which she grew up was the right fit for her. Everything felt fake after she and Quentin found a dead man when they were children. She felt that even though Agloe, New York, is a real "paper town" in the definitional sense, Orlando, Florida (where she and Quentin grew up), is the real paper town, because she doesn't consider Orlando authentic. She feels that at least the few small places that have been built in Agloe are authentic, so that is why she runs away and goes there.
Margo ran away, because, according to what she tells Quentin, the place for a "paper girl" is in a "paper town".
A paper town, as Quentin eventually finds out while searching for clues to where Margo might have gone, is a town that does not really exist. It is a point on a map put there by cartographers in order to enforce the copyright laws. (If another cartographer makes a map with the same - non-existent - town on it, it will be clear that the map was copied.) Margo ran way to Agloe, NY, which can be found on some maps but is not a real town.
By calling herself a "paper girl", Margo seems to imply that she does not really exist, that she is not "real." Quentin, again, during his quest to find Margo, finds out a lot of things about her that he had not known. He realizes that Margo had led two - or more - separate lives: she was one person for her friends, while being a completely different person in private. He comes to understand that Margo needs time alone to find out who she really is and who she wants to be.
Margo tells Quentin that originally she planned to leave after the school year was over, once she has graduated from high school. However, when she found out that her boyfriend was cheating on her, and that her best friend knew but did not tell her, she was compelled to leave earlier.