Emulsions are funny things. Most important: they change the texture of a mixture.
An emulsion is a mixture of two or more liquids that does not separate (or separates only very slowly). Visualize what happens when you pour oil and vinegar into a glass? After only a short time, the oil will float to the top and the vinegar will be at the bottom.
In an emulsion, tiny droplets, each the size of 100 nanometer or less, of one of the liquids (the dispersed phase) are suspended and distributed evenly in the other liquid (the continuous phase).
Most emulsions are not completely stable. They slowly separate into their composite liquids. This speed of separation is influenced by the starting quality of the emulsion, but temperature is also known to have a negative effect. In food industry, stabilizers are often used to keep liquids emulsified. If used, you will find stabilizers listed on the food label (either by its name, such as pectin or gum, or by its E number, generally in the range of E-400 to E-499).
Interestingly, no chemical reaction happens in an emulsion. However, an emulsion may have an effect on the taste of the liquid. This is related to the specific surface area of the dispersed liquid.
Emulsions are a subset of colloids. Milk is a good example of a colloid: tiny particles of fat and other ingredients are "floating" in water, never to settle. We talk about emulsions when all materials involved are liquid.
Sometimes two liquids will need some help in order to emulsify. Generally this helper liquid will reduce the surface tension of one of the liquids or stabilize.
Emulsions do not come into existence by itself; generally speaking it needs a lot of agitation, like shaking or whisking, for two liquids to form an emulsion.
In the kitchen, you hope to see an emulsion develop when making mayonnaise (the oil being "absorbed" by the egg yoke mixture) or when making a vinaigrette. Pro tip from James Martin when making a vinaigrette with oil and vinegar: add a tablespoon of (cold) water and whisk vigorously!