Mildew is a type of mould that grows in wet environments such as bathrooms. When common mould spores, which are almost always present in the air, land on a wet surface, they start to grow. Synthetic materials are somewhat resistant to mould, but once they get dirty, the mould has a food source and will start growing across the surface. Keeping your shower curtain cleaned will prevent the mould from getting a start. Once mould has made its presence, clean it quickly to stop it from spreading.
Launder the shower curtain using washing powder so the detergent can wash away any mould spores on the polyester. Use warm water so the curtain stays soft during washing.
Add a cup of bleach to the machine using the bleach dispenser so any stains remaining on the curtain are bleached out. Use safety precautions as listed on the bleach container when working with bleach.
Rinse the shower curtain in the regular rinse cycle of the washing machine, but add a cup of white vinegar to stop any further corrosive action of the bleach on the polyester.
Dry the shower curtain outside in the sunshine, if possible, so any mould spores in the air don't land on the curtain and start growing again. Wash the curtain at least once a week to remove the dirt the mould grows on, preventing it from spreading over the surface.