Shower curtains come in a variety of materials, a common one being polyester. Since shower curtains are regularly in contact with water, they become susceptible to stains, whether by a result of mineral deposits from hard water or even by mold growth. Polyester requires special care, as it's made from petroleum-based fibers that can melt under high temperatures. While it takes special care to clean polyester, it doesn't necessarily mean it takes a lot of work.
Take the polyester shower curtain off the curtain rack.
Put the shower curtain in the washer. Wash the curtain with cold water on the delicate setting. You can use normal washer detergent as the cleaning agent.
Dry the curtain at low heat on your dryer's "tumble dry" setting. If you have a clothesline, you can also let the shower curtain air dry outdoors.
Iron the curtain on low heat if you need to remove wrinkles. Be extremely cautious when ironing polyester, as it's easy to burn.