There are two kinds of emulsions, temporary and permanent. An example of a temporary emulsion is a simple vinaigrette. You combine the oil and vinegar in a jar, mix them up and they come together for a short time. Mayonnaise is an example of a permanent emulsion, consisting of egg yolks and oil. Egg yolks and oil would not naturally mix together, but by slowly whisking the oil into the egg yolks, the two liquids form a stable emulsion that won't separate.
Hollandaise sauce is another permanent emulsion, which is made of egg yolks and clarified butter.
Certain substances act as emulsifiers, which means they help the two liquids come together and stay together. In the case of mayonnaise and Hollandaise, it is the lecithin in the egg yolks that acts as the emulsifier. Lecithin, a fatty substance soluble in both fat and water, will readily combine with both the egg yolk and the oil or butter, essentially holding the two liquids together.
In a stable emulsion, what happens is that droplets of one of the liquids become evenly dispersed within the other liquid. The resulting liquid is thicker than the two original liquids were. In the case of salad dressing, oil droplets are suspended within the vinegar.
A fine powder also can help to stabilize an emulsion, and so can a starch. That's why roux is useful in thickening sauces. It's the starch in the flour that joins the butter to the liquid stock. A cornstarch slurry works the same way.
Other less obvious examples of emulsions are chocolate (an emulsion of milk and cocoa butter) and some sausages and forcemeats. Hot dogs are an example of an emulsion sausage where meat, fat and water are combined to form a smooth forcemeat which is then stuffed into a casing.