Purple/Light Blue (Default): Based on her character model from The Legend of Zelda: Twilight Princess. Masahiro Sakurai stated that Sheik's appearance reflects how she would have looked like if she appeared in that game, based on designs drafted up during that game's development.
Red: Based off Zelda's early appearances in The Legend of Zelda and Zelda II: Adventure of Link. This is also the only one of her color schemes in which she doesn't wear gloves.
Blue: Official art for A Link to the Past shows Zelda with a blue dress, while Sheik seems based off of Tetra with the blue costume and red scarf.
Pink: Based off Zelda's appearance in Melee as well as Ocarina of Time. It also makes her look like Princess Peach.
Green: If the original Zelda is beaten without a ring, Zelda's dress will be green.
Black: Based off Zelda's mourning outfit from Twilight Princess. Makes Sheik look like a ninja and Zelda look like Sherry Reinhart.