Egg whites are made almost entirely of protein. When we whip them, the motion of the whisk forces the proteins to unfold and recombine in rigid structures. Whisking transforms egg whites into a mass of foamy bubbles and the proteins form the bubble walls.
These proteins are pretty darn stable - until a smidge of yolk or a little oil gets involved. Egg yolk, fat, and even residual detergent on a bowl or whisk interfere with the protein structure of the egg whites. It's like replacing some key walls in a house of cards with jello: the structure just can't hold together.
You can usually still whip your egg whites if it's only a little bit of yolk or fat, but it will be harder work and will take you longer. Metal, glass, or ceramic bowls are best because their surface is smooth and easier to keep clean. Plastic can hold residual traces of both fat and detergent even after multiple washings.
Long story short? Keep those egg whites separated!
(Image: Emma Christensen)