Soap in Water

Soap molecules mix readily with water because they have a polar or ionic “head”, but the rest of the molecule is non-polar and will not mix readily with water. For a single molecule of soap there is no real option to this somewhat dissatisfying situation, however, when many soap molecules are mixed into the water the situation changes.

To use our analogy, soap is an alien (seriously, go back and read the analogy or this won't make sense). It has an end that is attractive to water and an end that is not. When a bunch of soap is mixed with water it clumps together in spheres, each soap with it's attractive head out and it's non-polar end inside, away from the water, like the picture here:

This is called a micelle.

The formation of micelles also allows effective mixing of water and non-polar things like oil.