Mayonnaise is an emulsion formed by carefully and slowly whisking oil and raw egg together. One type of mayonnaise uses whole eggs, but the more delicate and tasty kind of mayonnaise uses the yolk only. The other ingredients in a basic mayonnaise are small amounts of Dijon mustard, salt, pepper, and vinegar (or lemon juice). One egg yolk will "absorb" approximately 6 fl. oz. (175 ml) of vegetable oil. You can try to blend in even more oil, but you risk making the emulsion break down if you put in too much.
Some people say that the oil and egg must be at the same temperature, preferably room temperature, but I've never found that rule of thumb to mean much. You wouldn't, of course, use oil that has congealed because it's too cold. Anyway, I start by separating an egg and dropping the yolk in a bowl big enough to allow for easy and vigorous whisking. I dip the end of a small clean whisk (or a fork) into the jar and pull about about a teaspoon of mustard for one egg yolk. It's easier with a whisk, but you can also make the mayonnaise just using a fork. (By the way, in France you can buy Dijon mustard in tubes, as you can commercially made mayonnaise.)
Before starting to whisk the yolk and the mustard together, add pinches of salt and pepper plus a small splash of vinegar or lemon juice (three or four drops — no more than half a teaspoon). The mustard and vinegar will help the egg yolk "absorb" the oil and make the emulsion come together. La mayonnaise « prendra » plus facilement. If you want to add herbs or pureed garlic, you can put them in at this stage. Use whatever kind of vinegar you like, other than balsamic. Sometimes I use wine vinegar (white or red), or cider vinegar, or just plain distilled vinegar (vinaigre blanc).
While I don't worry much about temperature, I do believe that it helps to whisk together all the ingredients I've listed and let that mixture sit and rest for a few minutes before starting to incorporate the oil. Somehow, that seems to prepare the egg yolk to receive the oil. As far as oil goes, you can use what you like. I think huile de colza (canola) makes a thicker mayonnaise than does huile de tournesol (sunflower), but either of these widely used oils will work. So will olive oil, but unless it's the very best olive oil the flavor might turn out to be a little bitter. What I normally do is use a mixture of half colza or tournesol and half olive oil. Any neutral-tasting vegetable oil can go into such a blend.
So you have the egg, mustard, vinegar, and seasonings blended together in the bowl, and you've measured out the 6 fl. oz. of oil into a cup with a pour spout. Now comes the test. Start whipping the egg + mustard mixture with the whisk using one hand and then, using your other hand (or a helper), begin pouring the oil into the bowl in a very thin stream, or even drop by drop, as you whisk. You'll see the oil start blending with the egg into an emulsion that gets thicker and thicker as you go. As the mayonnaise develops, you can begin going a little faster in pouring in the oil. Don't stop whisking. If you accidentally pour in what seems like too much oil all at once, don't try to mix it in too fast. Instead use the whisk to carefully touch just the edge of the pool of oil and gradually pull it into the mayonnaise until it all smooths out. In the photo above, about half the oil has been incorporated, and there's a pretty big pool of oil on its surface. I'm carefully blending it in, little by little.
I made the mayonnaise in this last photo using a half and half blend of sunflower oil and olive oil, with white wine (Chardonnay) vinegar. The egg yolk, olive oil, and Dijon mustard all contribute to its yellow color. It is well emulsified (la mayonnaise a bien pris) but it's a soft, not stiff, emulsion. If your mayonnaise seems too stiff, carefully mix in a few drops of vinegar, lemon juice, or even water to thin it. By the way, the acid in the mixture (vinegar or lemon juice, plus mustard) keeps bacteria from growing and prevents the mayonnaise from going bad. It's best, of course, to keep it in the refrigerator if you aren't serving or eating it immediately. If you use olive oil in the mixture, however, it can congeal at low temperatures, and that might break the emulsion, causing the mayonnaise to separate. You can try to recoup it by bringing it back to room temperature, adding a few drops of vinegar or lemon juice, and whisking it back together.