The Physics of It
Oil and water don't mix on their own because water molecules are polar and oil molecules aren't. They separate. An emulsion is what happens when you force one liquid into tiny droplets and suspend them inside the other using a third ingredient: an emulsifier.
The terminology: the liquid broken into droplets is the dispersed phase, the liquid surrounding those droplets is the continuous phase, and the emulsifier sits at the boundary between them, holding the structure together.
Most kitchen emulsions are oil-in-water (the oil is dispersed in a water-based continuous phase). Mayonnaise, hollandaise, and vinaigrette are all oil-in-water. Butter is the reverse: water droplets dispersed in fat. When you brown butter, you're cooking off that dispersed water.
Emulsifiers: Lecithin and Mustard
An emulsifier works because its molecules have a hydrophilic (water-attracting) end and a hydrophobic (fat-attracting) end. They coat the droplets of the dispersed phase and prevent them from coalescing back together.
Egg yolk is the most common kitchen emulsifier. The active compound is lecithin, a phospholipid present at roughly 9-10% of the yolk's weight. One egg yolk contains enough lecithin to emulsify about 240ml of oil (1 cup), though most recipes use less oil than that maximum to maintain a stable, creamy texture. You can push one yolk to 300ml of oil, but the emulsion gets thinner and less stable.
Mustard contains mucilage compounds that act as a supplementary emulsifier. It doesn't work nearly as well as lecithin on its own, but in combination with egg yolk (as in many aioli and vinaigrette recipes) it adds stability and flavor. In a vinaigrette without egg, a teaspoon of Dijon helps hold the emulsion together temporarily.
Mayonnaise: Ratios and Why It Breaks
A stable mayo is roughly 1 egg yolk to 120-180ml of neutral oil, plus an acid (lemon juice or vinegar) and salt. The acid helps with flavor but also slightly thins the continuous phase, which makes it easier for the lecithin to do its job.
The most common failure mode is adding oil too fast. If you dump oil in before the emulsification process has started, you overwhelm the available lecithin before it can coat the droplets. Add the first tablespoon very slowly, whisking constantly. Once the emulsion takes hold and the mixture thickens, you can add oil faster.
When mayo breaks (it separates into an oily mess): start with a fresh yolk in a clean bowl. Add the broken mayo drop by drop to the fresh yolk, whisking constantly. You're using the new yolk as an emulsifier base and slowly re-dispersing the broken mixture into it. Works reliably.
If your mayo is thin but not broken, it usually means you're under-emulsified: not enough lecithin for the amount of oil. Add another yolk and whisk it through.
Vinaigrettes: Temporary vs. Stable
A standard vinaigrette is roughly 3 parts oil to 1 part acid. You can shake it in a jar and it'll hold for maybe 30 seconds before it separates again. That's a temporary emulsion: you've broken the oil into small droplets, but there's no emulsifier holding them there.
Add a teaspoon of Dijon and you get a semi-stable emulsion that holds for several minutes. Add a small amount of honey or tahini and you get similar results. None of these are truly stable emulsions like mayo. They'll all separate eventually. That's fine for a salad dressing you're using right away. If you want something shelf-stable, you need more lecithin (an egg yolk) or a commercial emulsifier.
Hollandaise and Heat
Hollandaise is an egg yolk emulsion with clarified butter, but it has an additional complication: temperature. The egg yolks need gentle heat to begin setting (which thickens the sauce and helps stabilize the emulsion), but too much heat scrambles them.
The working range is roughly 60-70°C. Below that, the yolks aren't contributing much structure. Above 75°C, you start getting scrambled eggs.
Make it over a double boiler: whisk yolks with a small amount of water or lemon reduction over barely simmering water, keeping the bowl from touching the water. Whisk until the mixture is pale, thick, and holds a ribbon when you lift the whisk. Then add warm clarified butter slowly, same technique as mayo. Final sauce should hold at 55-60°C.
When hollandaise breaks: if it's still warm, whisk in a small amount of cold water or ice water to bring the temperature down and re-disperse the fat. If it's too far gone, start a fresh yolk reduction and add the broken sauce slowly.
The key difference from mayo is that you can't fix it at high heat. The eggs cook. Work gently, keep your temperatures controlled, and hollandaise is not that difficult.