An emulsion is a suspension of two things that normally separate.
Mayonnaise and Hollandaise are examples, they're vegetable oil
suspended in water, and oil and water are famous for not getting
along. Vinegar and oil dressing is another example.
To keep them from separating, you stabilize them with an emulsifier.
An emulsifier has hydroscopic and liposcopic properties, so it absorbs
both water and fat and keeps them stuck together. The most common is
lecithin, found in egg yolks and mustard seed. Mayonnaise and
Hollandaise are both made with egg yolk, mayonnaise generally also has
ground mustard seed.
Most sauces using these also involve acid in the form of vinegar or
lemon juice. The acid helps to denature the proteins and stretch them
out, allowing their hydro/liposcoping surfaces to come in contact with
more of the water and fat in question, thus helping to stabilize the
sauce.
I looked around the classroom in the middle of lecture and oh god, the
glazed-over eyes. To me it's just delicious science.