This has been touched on a couple times in some other threads, so I figured we could give it it's own separate discussion here.
1. Darth Vader is Luke Skywalker's father
But in Star Wars (1977), Obi Wan Kenobi and Uncle Owen both explicitly tell Luke that his father is dead, and Obi Wan explicitly tells Luke that his father was betrayed and murdered by Obi Wan's former pupil Darth Vader. When those lines were written by George Lucas in the mid-70's, was he already committed to the idea that Obi Wan is lying, with the rationalization that Obi Wan has a "certain point of view"?
In fact further research shows even more reason to doubt that Vader was Luke's father before Empire Strikes Back was released. Not only does Star Wars make Vader and Luke's father seem like two distinct characters, but the first draft of Empire Strikes Back's screenplay, which Leigh Brackett wrote based on George Lucas's story treatment, has the ghost of Luke's father appearing to Luke to instruct Luke. Then Lucas wrote the second draft and it was in this second draft that Vader is first said to be Luke's father. Michael Kaminski argues in his book that the twist didn't exist before then; if someone wants to say that it did, they bear the burden of proof.
2. Leia is Luke's sister
But in Star Wars (1977) Luke very clearly has an attraction for her, which causes tension between him and Han in a scene on the Millennium Falcon. Then in Empire Strikes Back, Leia kisses Luke pretty hard in front of Han, and Luke's reaction clearly shows he was digging it in a non-brotherly way. And the Blu-Ray shows deleted footage from that scene where Luke and Leia share an intensely romantic exchange (this time not to spite Han but actually from the heart). All of this makes sense if you believe Gary Kurtz, the producer who left the series after Empire Strikes Back, when he admitted in an interview that Luke's sister, the "other skywalker" was originally not going to be Leia but was going to be a new character to be introduced in the sequel trilogy (episodes vii-ix). Kurtz opposed Lucas's last minute decision to resolve the love triangle in Return of the Jedi by switching the sister to Leia. Supposedly Lucas wasn't sure at that point if episodes vii-ix would ever get made so he decided to rush the resolution of the "other skywalker" subplot. You have to admit, Return of the Jedi's "reveal" of Luke and Leia's relationship feels a bit shoehorned in....
3. Darth is Vader's first name?
In Star Wars (1977) Obi Wan refers to Vader before he turned to the dark side as "a young Jedi 'NAMED' Darth Vader...." Then when he fights Vader on the Death Star, he says "Only a master of evil, Darth...." and "You can't win Darth..." For those who try to argue that this evidence is weak and inconclusive, how do you explain the following:
This way of speaking is never repeated in a subsequent Star Wars movie. Never again is someone said to be "named" Darth so-and-so. And never again is someone referred to merely as "Darth".
In fact, in the first draft of The Star Wars, Darth Vader is General Darth Vader, so "general" is his title, yet he still had the name "Darth."
4. Did Anakin's wife die in childbirth?
Even though Luke says he has no memory of his mother because he never knew her, Leia contrasts with this by saying she has a little bit of memory of her mother because she died when Leia was very young. This conversation clearly contrasts Luke and Leia where there mother is concerned; they did not know their mother for the same length of time.
Further research also corroborates this. In the story conferences between Lucas, Kasdan, Marquand, etc. for Return of the Jedi, Lucas elaborates on the backstory as it was in his mind at that time, and explicitly states that Anakin did not know his wife was pregnant, she kept it a secret from him because of the changes that have been taking place in him as he's drawn more and more to the dark side and becomes less and less the Anakin she's familiar with. She confides about the pregnancy to Obi Wan and gives birth in secret, and survives the birth. She takes Leia to Alderaan, where she dies while Leia is two or three years old.
Based on that story conference, dialogue was written for the scene on Dagobah in Jedi after Yoda's death when Obi Wan appears and discusses backstory with Luke. The scene as scripted goes longer than in the movie; Obi Wan explains some of the details outlined above, and even says that Luke's Uncle Owen was Obi Wan's brother. This full conversation between Luke and Obi Wan appears in the novelization of Jedi, even though it was cut from the movie.
When Luke decides to reveal to Leia that he is her brother on Endor, there are multiple ways he could choose to start the conversation. On the basis of the information that Obi Wan revealed to him earlier (that Luke was taken from his mother at birth while Leia went with their mother to Alderaan), he decides to ask about her memories of their common mother, memories that he wouldn't have, but that she would have because she wasn't separated at birth the way he was.
Explanations by fans on the internet, that Leia "uses the force" to have image memories of her mother, are just feeble attempts to reconcile something that in reality is just George Lucas changing his mind when it came time to write Revenge of the Sith.