Add everything except the oil to a food processor or blender. Turn the machine on and, with it running, VERY slowly add the oil. (VERY slowly. We're talking drop by drop, or in such a narrow stream it's almost by the drop.)
Within a few minutes, the mixture should emulsify.
Mayo can be refrigerated for up to a week.
Notes
The key to a successful emulsion -- which mayonnaise is -- is to add the oil very, VERY slowly. You're essentially making oil and "water" mix, so the oil has to be able to be broken into tiny droplets.Some people have had success emulsifying mayonnaise by blending it with an immersion blender. I don't have one, so I can't vouch for that.