For more detailed instructions, see this illustrated tutorial on how to make mayonnaise.
Also, here's a tutorial on how to separate eggs.
A light, neutral-flavored oil like safflower, canola or soybean oil will give the best results, but any blended oil labeled "vegetable oil" or "salad oil" will do the trick.
Finally, it can't be said too often: For safety reasons, use pasteurized eggs for this and any other preparation that contains raw eggs.
Let all your ingredients come to room temperature before you begin.Using an electric hand mixer, stand mixer or a wire whisk, whip the egg yolks for a minute or two, until they're thoroughly beaten.
Add the 2 tsp vinegar and whisk for about half a minute. Then add the salt, and the cayenne if you're using it, and beat for another 30 seconds or so.
Now, with the mixer going full speed (or with your arm whisking as hard as it can) add the oil very slowly, as little as a drop at a time.
When the emulsion starts to form, you can add the oil more quickly, but keep it at a fairly moderate stream. Adding the oil too fast will break your mayonnaise.
When the emulsion thickens, add a teaspoon or so of vinegar to thin it out. Continue adding oil, stopping occasionally to add more vinegar if the mixture gets too thick.
Finish with the lemon juice, both to add a bit of tang as well as to achieve the right consistency.
Store unused portion in the refrigerator, where it should keep for a day or two.Makes 1 pint of mayonnaise.