Emulsion

Chemically, emulsion is the process of creating bonds between liquids that don't mix well into something that's more stable. Blah blah.

In terms of cooking, emulsion is effectively a process of making oil and water mix together, so that the result is not a greasy mess, but a creamy sauce or stock.

I've never tried to emulsify anything, so take all this with a grain of salt. I've mostly seen some sort of oil (or fat) emulsified with water, so I'll talk about that. There's generally some stabilizer that can bond itself with both oil and water to make this happen. Generally, the stabilizer, oil, and water, are vigorously mixed together until they all become a happy mixture. I think mixing too vigorously can cause the emulsion to break and the oil and water are no longer a happy mixture.

Pasta sauces

Good pasta sauces are nice and creamy and cling to the noodles. They shouldn't be greasy. However, it's generally made up of cheese (which has lots of oils) and water (tomato sauce, vodka sauce, etc) which don't mix too well. Chefs always say to reserve some pasta cooking liquid because the liquid has a lot of starch from cooking the pasta noodles. This starch is what helps bind the sauce together as the cheese melts.

I pretty much never make pasta, but the idea that the starchy pasta water can act as an emulsifier seems like a pretty important idea to keep in mind for other emulsions.

Tonkotsu

Tonkotsu broth is a huge mix of fat from pork bones, chicken bones, and water. It's emulsified, and is super creamy and rich. Without the emulsion it would just be one layer of pork and chicken fat, and one layer of stock. The emulsion happens because of how long the broth is boiled. The gelatin in the pork and chicken bones is drawn out in the cooking process and acts as a stabilizer, and the boiling broth acts as "mixing" to emulsify the fat and water together.

Mayonnaise

Mayonnaise is just a stabilized mixture of eggs, oil, some lemon juice, and optionally garlic, salt, pepper, or other flavorings. The stabilizer I've seen used is some mustard, and the mixing is done with an immersion blender as the oil is slowly drizzled into the mixture.