Oil and water refuse to mix — until you break the oil into droplets so small they stay suspended, ideally with an emulsifier standing guard around each one. Egg yolk's lecithin, mustard, honey, and even pasta water's starch are all emulsifiers.
Every "creamy" sauce that contains no cream is an emulsion: mayonnaise, hollandaise, beurre blanc, aglio e olio, a proper vinaigrette. When a sauce "breaks," the droplets have merged back into an oily slick — usually from too much fat added too fast, or too much heat.
The rescue is always the same idea: start a new emulsion and slowly whisk the broken one into it. For mayo, a fresh yolk; for a pan sauce, a splash of water and a whisk off the heat.
Start with the water phase and the emulsifier (vinegar + mustard; yolk + lemon).
Add fat in a thin stream — drop by drop at first — whisking constantly.
Loosen with water if it gets too thick; thickness means the emulsion is crowded, not broken.
Season last: emulsions mute salt and acid.