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.
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.
Johnny Depp wore contact lenses, which served as sunglasses, so he wouldn't be squinting in the sun all the time.
(at around 1h 30 mins) Jack (the monkey) is seen smiling when Barbossa thanks him specifically after he returns the Aztec coin to him, after Jack Sparrow chases him. In the special features on the DVD, it is revealed that the monkey did this purely by chance, and it was not planned out in any way.
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.