In Episode VI: Return of the Jedi, Luke asks Leia if she remembers her real mother. Leia replies that her mother died when she was very young, but she was "very beautiful. Kind, but...sad." After Episode III: Revenge of the Sith came out, however, fans questioned how Leia could remember her mother when Padmé died in childbirth.
Leia's lines about her mother made more sense at the time Return of the Jedi came out, since the backstory about Padmé and Anakin Skywalker had not yet been established. In the novelization of Return of the Jedi, written by James Kahn, Obi-Wan Kenobi tells Luke that Anakin didn't know his lover was pregnant when he became Darth Vader, and Obi-Wan hid her away to protect her. After she gave birth, Obi-Wan took Luke to Tatooine and she took Leia to Alderaan.
This and other parts of the novelization, however, clearly contradict later sources. In addition to giving a different explanation for the twins' birth, the book states that Owen Lars was Obi-Wan's brother. Therefore, the original explanation for how Leia remembers her mother no longer works.
Some fans have speculated that Leia did not remember her real mother, but rather, Bail Organa's wife, Queen Breha. In Children of the Jedi by Barbara Hambly, Leia mentions being raised by her aunts, suggesting that her adoptive mother died when she was young. However, Star Wars: The Annotated Screenplays states that George Lucas intended for Leia to remember her real mother, and Leia's Star Wars Databank entry states that her memories are of Padmé.
How is it possible for Leia to remember Padmé after meeting her as a newborn, for only a few seconds? Patricia C. Wrede's novelization of Episode III describes the newborn Leia as looking around her, intent to memorize every detail. Perhaps Leia's Force-sensitivity allowed her to form memories even at such a young age. Leia's Force-sensitivity is more personal and emotional than Luke's; her first hint of Force ability, in Episode V: The Empire Strikes Back, is sensing Luke on Bespin. Even as a newborn, her instinctual connection to the Force could have helped her form a bond with Padmé.
It's also possible that Leia gleaned images and impressions of her mother through the Force even after Padmé's death. As Yoda tells Luke in The Empire Strikes Back: "Through the Force, things you will see. Other places. The future...the past...old friends long gone." Even though Leia has no formal Jedi training until after Return of the Jedi, she could have learned about her mother through visions in the Force, which she then mistook for memories.