Hey, It's That Voice!: Listen to the voice of the Imperial officer who tells Vader that there's no one on board the Millennium Falcon. Are you suddenly thinking of The Simpsons? That's because several of the British incidental cast were dubbed by none other than Harry Shearer. You can also hear him dubbing the guy on the Star Destroyer who sees the escape pod ("There goes another one") and one of the Rebel pilots in the Death Star attack ("The guns...they've stopped!")
Though it's hard to tell because of the heavy editing, Garindan (the Imperial Spy in Mos Eisley) is voiced by John Wayne.
It Will Never Catch On: Out-of-universe, George Lucas really had a lot of threats from Fox to pull the plug on his "space movie".
Although it did mean Lucas got full sequel and merchandising rights easily. He waived the money he would be paid as the director, and the studio believed they would just be losing less money thanks to that.
Method Acting: When Han was ad libbing his response to security over the comm, Harrison Ford only skimmed over the script and instead ad libbed Han's ad libbing, to make it seem more genuinely spontaneous.
Mis-blamed: Contrary to popular belief, Greedo shooting first was not Lucas' fault, the MPAA insisted he put it in there in order for the movie to keep its PG rating.
No Budget: Started on an 8 million budget (3 mil. for the live action, 5 million for the special effects) but ballooned to 11 million due to the many production problems and reshoots, including blowing half the FX budget on four shots that George considered unacceptable.
The scene of Han trying to talk the Stormtroopers out of investigating the shootout they've just had. Depending on who you talk to, Harrison Ford forgot his lines, never read them at all, or just learned them shortly before shooting. He even cringes when he realizes how he sounds.
"I can't see a thing in this helmet!"
The stormtrooper on the right smacking his head on the low doorway. Instead of trying to edit it or even ignore it in later releases, they in fact did the exact opposite by lampshading it with a very audible "thunk" as it happens.
Particularly infamous example. Apparently, Mark Hamill, Harrison Ford, and Carrie Fisher were all really wierded out by Lucas's decided lack of good dialogue skills. They stood up to him—and Lucas, chastened, allowed the actors to basically improvise their own wording for the basic points of the screenplay's dialogue.
Troubled Production: Like you cannot imagine; making the film was such a terrible experience for George Lucas, that he left directing the next two films in other hands, and wouldn't direct another movie until The Phantom Menace many years later.
During initial drafts, Darth Vader's iconic suit would have been a spacesuit as he needed to board the Tantive IV through space. It ended up since rewritten to being a permanent life-support system.
The role of Obi-Wan Kenobi was originally written with Toshiro Mifune in mind. Depending on who you talk to, either 20th Century Fox wasn't keen on giving Mifune another whirl (although Mifune could speak English, all productions where he was speaking English ended up dubbing over his voice due to his thick Japanese accent) or Mifune wasn't available. Either way, the role went instead to Alec Guinness.
Peter Cushing was considered to star as Obi-Wan Kenobi before being cast as Grand Moff Tarkin.
Originally, Sissy Spacek was supposed to be cast in the role of Leia, while Carrie Fisher was supposed to be cast in the lead role of Carrie. However, after Fisher objected to doing a nude scene, the two actresses swapped roles.
The prologue of the tie-in novel outlined a largely different backstory than what the films eventually portrayed. In particular was that it said that Emperor Palpatine was originally a helpless puppet of the Imperial bureaucrats rather than the ultimate string-puller that the later films revealed him to be.