What you will need:
full cream milk
An improved procedure [source]
1. Pour some full cream milk into a dinner plate
2. Squeeze some drops of food dye into the milk (one colour on one side, other colour on other side)
3. Add a drop of detergent onto the edge of the plate so it runs down into the milk
4. The colours begin swirling like a psychadelic pattern
5. You can add more detergent if the motion stops
Explanation [The explanation in the above video is incorrect]
Like milk itself, this classic experiment is actually quite complicated. Milk is a suspension of tiny fat globules (about 4% by volume) in water, plus a whole range of proteins, sugar (lactose) and nutrients like calcium. Detergent is a form of surfactant (short for surface active agent) - individual detergent molecules can bind with both water and oil.
The swirling effect in milk is probably driven by the detergent molecules racing around and coating the fat globules. As the detergent molecules are "consumed" by the fat, they create currents. You'll notice colours from the opposite edge of the plate appearing near the detergent and then shooting across the surface.