When you do laundry what causes nonpolar oil or greasy dirt to mix with the polar water?

Most laundry dirt is oily or greasy. Detergents can mix with both oil and water, so when the wash water goes down the drain, the soap and dirt go with it.

Detergent molecules are designed with a polar head of the molecule and a non-polar tail. The detergent molecule thus "ties together" the water and the grease. Soap performs the same function.