Mayonnaise is the ultimate example of emulsification: just a few grams of egg yolk can "glue" hundreds of milliliters of oil and a few tablespoons of water into a thick, creamy emulsion. Lecithin in the yolk is the natural emulsifier, and prolonged whisking breaks the oil droplets down to micron-sized particles, making them unable to coalesce.
The most common mistake when making mayo is adding oil too quickly—you must start drop by drop. Once the sauce thickens, warm water becomes a secret weapon: add a few drops and whisk; the emulsion will instantly become smooth again. Crucially, temperatures above 40°C (104°F) will break the emulsion (causing it to "split"), so mayo cannot be heated. Its extended family is vast: add minced garlic for aioli, tomato paste for Thousand Island dressing, or fresh herbs for green sauce—master the basic recipe, and you can create endless variations.