The best thing to wear to a Moulin Rouge party is a costume in a style from the Edwardian time period. Because the movie takes place in the nightlife scene, fancy or exciting clothing is appropriate.
A fun outfit for a woman to wear to a Moulin Rouge-themed party would be a courtesan costume. As About.com notes, this could be as modest or as revealing as the wearer likes, but would probably include some basic elements such as a tightly-fitted corset or bodice, lacy tights, and high-heeled shoes. A skirt is optional. A small, feminine top hat, gloves, and flashy jewelry would complete the look nicely. For the hair, a half-up, half-down look with lots of curls and volume is an appropriate choice.
For a man, a sleek black suit with a couple of inventive details would work. White tie and tails would look the most striking. Pops of color throughout the outfit, such as a bright red vest or coat, would set a tuxedo apart as something special. According to About.com, gloves are a must, as is a top hat. Dapper shoes and perhaps a cane complete the look. The face should be clean-shaven, except perhaps for a neat mustache, and the hair should be worn short.