Beam Me Up, Scotty!: Anakin's line about sand is "I don't like sand. It's coarse and rough and irritating and it gets everywhere. Not like here. Here everything is soft and smooth." It's often misquoted as "...Not like you"; he may have been thinking of Padmé, but he didn't actually say it so openly.
Deleted Scene: In which Padme completes Threepio by putting on his shiny gold skin while Owen Lars watches. Considering this would have raised questions as to why Owen doesn't recognise Threepio in A New Hope, it's no wonder it got scrapped.
Separated-at-Birth Casting: In certain shots, Hayden Christensen looks a lot like Mark Hamill (In Episode III, when his hair grows out, the resemblance is closer. Joel Edgerton was cast because he looked like a younger version of Phil Brown, Owen lars original actor. Jack Thompson as Cliegg Lars, his father, also shares strong traits.
The entire Aggressive Negotiations conversation during the dinner scene between Anakin and Padme was ad-libbed by Hayden Christensen and Natalie Portman at George Lucas' request, due to his not being happy with the romantic dialogue he wrote for that scene.
Anakin's silhouette resembling Darth Vader just before he goes off to find his mother was a complete coincidence, and wasn't edited in.
Jedi Master Sifo-Dyas was originally just a flimsy alias for Darth Sidious known as Sido-Dyas, but a typo was made in the script. Lucas preferred the new name, and the plot point about him was changed to make him an actual Jedi that had disappeared.
A deleted scene shows Anakin and Padme's sentencing before the arena scene. It reveals they aren't being killed for opposing the new Confederacy, they're being executed for murdering all those innocent workers and security in the factory before they can use their political clout to squirm out of responsibility.
The novelization of Return of the Jedi states that Owen Lars was Obi-Wan's brother, instead of Anakin's stepbrother as finally shown in the film.
Sidious's new apprentice was originally going to be female, the character even got to the design phase, before being swapped for Dooku. These designs would go on and become the basis for Asajj Ventress' character in the EU, from where she was lifted into a major supporting character in Star Wars: Clone Wars and Star Wars: The Clone Wars.
Jedi Master Sifo-Dyas was supposed to be an alias for Darth Sidious known as Sido-Dyas instead of an actual Jedi. Apparently a typo in the script lead to them developing Sifo-Dyas as a real Jedi and turning it into something of a Deceased Fall-Guy Gambit.
Ex'Chancellor Valorum was originally planned to make a cameo in the film, but he was written out when Terance Stamp refused to reprise the role due to disliking his experience working on The Phantom Menace.
This is the first Star Wars movie that did not end up being the highest-grossing movie of the year it was released in, losing that title to both Spider-Man and The Lord Of The Rings: The Two Towers. It's also the only one to not accomplish this feat, provided that you don't count the Pilot Movie of The Clone Wars (which was not originally intended to be released theatrically).
To date the only Star Wars film where the camera pans up rather than down after the opening crawl.