According to the DVD commentaries, Geoffrey Rush has a theory that people watch the screen from left to right, just like when they read a book. Therefore, he tried to be in the left side of the screen as often as possible. He was particularly intent on doing this in the scenes with the monkey and Keira Knightley, because he didn't think anyone would look at him otherwise.
Johnny Depp wore contact lenses, which served as sunglasses, so he wouldn't be squinting in the sun all the time.
Clothing and smears of charcoal were used to conceal Johnny Depp's numerous tattoos. The "Jack Sparrow" tattoo on his arm in the movie is a fake, but he got a real replica after finishing the film, in honor of his son Jack.
There is a small scab on Jack Sparrow's jawline that gets bigger and bigger throughout the movie. The make-up artist and Johnny Depp did it on purpose as a prank.
According to the screenwriters' commentary on the DVD, Will Turner is the best swordsman in the film, Barbossa and Commodore Norrington are evenly matched, and Jack Sparrow is actually the worst.