How can I make my own mayonnaise properly?
The oil needs to be slowly added to the egg, vinegar, and lemon juice base while being vigorously whisked. Only add more when the previous amount has been fully incorporated. I generally add the oil 1/4 cup at a time, dripping it very slowly into the base. Once the initial 1/4 cup has been added and emulsified it becomes a lot easier to incorporate the rest, and you can add it slightly faster, but still 1/4 cup at a time.

This is vastly simpler in a food processor. I have a Cuisinart 7-Cup Food Processor which has a little white pusher piece with a drip-hole in it. You can simply add the oil into this 1/4 cup at a time, it will slowly drip into the mix. Use the metal blade attachment.