This is a base-line aioli recipe that I have used in restaurants:
1 egg yolk
~1/2 clove garlic
1 cup oil
On emulsification: There are a few different kinds of emulsification, achieved via different emulsifiers. The compounds in egg yolks are what we would consider true, or chemical emulsifiers, as they actually bond to fat molecules, keeping them separate from one another in a matrix of water. When the fat in your mixture does all run together, it is "broken," meaning that you have a puddle of liquid with a big oil slick on top; not aioli. As oil exhibits cohesive properties, it does not want to be parted from itself once it has been allowing to join together, so the key, and the purpose of the slow addition of the oil, is to imagine that you are in fact lubricating the oil molecules at all times with a surrounding layer of water. Fail to keep the oil "immersed," and you are done for.
Sciency scientist types have found that a single egg yolk is capable of emulsifying gallons of oil. The true limiting factor for the stability of your emulsion is the ratio of water to fat; no number of yolks will emulsify more than a certain quantity of fat into a given amount of water. Thus, we start the aioli both with some of the lemon you will be using for flavor, and an additional quantity of water for insurance. While it is possible to emulsify 1 cup of oil into a bit of lemon juice with a single yolk, this typically results in an extremely stiff aioli, which is somewhat unpleasant on the palate, so you will want water anyway for consistency purposes. The basic ratio of this recipe is simple: 1 yolk, 1 cup oil, ~1-2 TB liquid (none of which necessarily need be lemon juice)
On oil: Say you smear 2 ounces of aioli on your sandwhich. That means you have about 1.5 ounces of straight oil that you are about to eat. If you use nothing but extra virgin oil, you have pretty much just ruined your sandwhich with the addition of a jigger of strongly flavored oil; hence, aioli tends not to be made with 100% EVOO. I have encountered two schools of thought on this; 1: using a blend of extra virgin olive oil and something more inert, typically canola oil; 2: using 100% olive oil, but something mildly flavored, meaning either pure olive oil or a super light EVOO, or a blend. In the former, I would go something like 60:40 canola:EVOO, assuming your EVOO is pretty good. In the latter, try to find a decent pure olive oil (which is to say an oil which is 100% derived from olives, but does not have a low enough oleic acid content to qualify as virgin/extra virgin, but is also not shitty enough to be classified as pomace oil).
Crush the garlic into a paste in a mortar or thoroughly smash it with the side of a knife, a saute pan, etc. It needs to be completely obliterated. Generally, you start your aioli with a little bit of garlic as it is a natural emulsifier, but you want to leave some aside to adjust so it doesn't turn into a garlic bomb. Generally, if I am making a 3 yolk batch, I would smash up 4-5 cloves of garlic, and ultimately end up using about half of that.
Combine the egg yolk, your starter garlic, a squeeze of lemon juice, and a splash of water in a bowl. Whisk the mixture until fully combined, and the egg yolk begins to lighten in color and soften in texture. This indicates that you have begun to denature the proteins in the yolk, which frees up the emulsifying compounds.
Begin adding oil whilst whisking vigorously. You really need to beat the shit out of it for the first third or so of your oil. Keep it moving, and really use the space in your bowl so that at no point is any one bit of the aioli in contact with a significant quantity of straight oil. This stage will take the majority of the time, say 6-10 minutes for a 1 yolk batch.
If all has gone according to plan, you should have a stiff, stable emulsion with 1/3 to 1/2 of the total oil added. Now is a good time to adjust for liquid, knowing that it will increase in stiffness as you add more oil. This done, you can add your fat much more quickly, provided your whisking arm isn't too tired to keep things in motion.
When finished, adjust for salt (way more than you think you need), garlic (way more than you think you need), lemon (not as much as you think you need) and additional seasonings (anchovy, pimenton, calabrian chile, seville orange zest, etc). It definitely needs to age a couple hours to mellow, but you should know that as a rule, aioli is thrown away at the end of every night, in every restaurant.
Whisk vs food processor: Keeping in mind that in Italy and France aioli is traditionally made start-to-finish with a mortar and pestle (not to mention that it is often solely assembled using the power of the garlic itself, and egg yolk is considered by some nonnas to be a cheat), a whisk actually starts to feel luxurious. In anything less than a 4-5 cup batch (which is a ton of aioli) I have found that a food processor is actually highly at risk for breaking the emulsion, as the addition of oil is less consistent, and the motor heats the mixture, thereby precipitating additional caution which can actually result in it taking longer. With experience, you should be able to throw together a batch of aioli sufficient for a single meal in less than 10 minutes, without causing a big annoying mess in your cuisinart.
Broken aioli? Just add water! Put some liquid in a new bowl, and begin by whisking a tiny bit of your broken oily mess into that water. It should form up into a loose but emulsified sauce. You can now proceed even more slowly than you did before, because hurrying is probably how you broke that shit in the first place, you slouch. Once your arm starts to fall off, you can really appreciate the merits of doing things right the first time.