1 egg yolk (or more depending on how big a batch you need- 4 yolks makes about a quart of mayo)
oil of choice (I used olive)
vinegar or lemon juice (only a drop or two)
whisk or mixer (mixer highly recommended, as your arm will likely fall off whisking this much)
One yolk, in the mixer. Right off the bat, I want you to notice how orange our egg yolk is (you can read more about that here). This is a contributing factor in our final product NOT being fluffy white in the end (plus the fact that we used olive oil).
The gist of it is that you're going to slowly add the oil into the mixer. And I do mean slowly. Start with a drop at a time. I'm not kidding. If you add too much at a time it wont turn out. You're going to need to exercise some patience on this one. Add in a drop or two of white vinegar or lemon juice to help stabilize your concoction. Wait until each addition is completely incorporated before you add any more. Whisk as fast as your arm (or your mixer) can.
Still just a drop at a time... patience my dear. It'll be worth it! If things are starting to look a little dry, you can add A drop more of vinegar or lemon juice. Trust me, one drop makes a big difference. Continue on slowly adding in your oil. Once it's started to look creamy you can add a teensy bit more at a time, but still take it slow. After all this work, you don't want to wreck it now! Expect that this whole adding the oil part will take you ten minutes or so.
Aim to add no more than a cup of oil per yolk. Any more and you're going to have a mess on your hands. Season your mayo with some salt, maybe pepper and a splash of lemon juice to taste. Or if you're making tartar sauce, you can find the details on how to make that here.
We found that one yolk made just about enough tartar sauce for the two of us. But I do this repulsive thing where I mix my tartar sauce up with ketchup for my fish, so any normal human being may need to make more tartar sauce than we did. It'll only last for a few days in the fridge, so don't bother making more than you're going to actually use.
Much thanks to this site for teaching me how to make mayo! We may very well never go back...