What's the difference between hollandaise, mayonnaise, and aïoli?

Mayo, at its most basic, is egg yolk and oil, with a little vinegar, whipped into an emulsion.

Aioli starts with oil and garlic, and sometimes vinegar or lemon. Some versions (French-Provençal, apparently) add egg yolk for an end result close to mayonnaise, but the yolk is not required.

Hollandaise is a cooked sauce made from egg yolk and butter, sometimes flavored with lemon and pepper. I think it must have a much larger ratio of egg yolk to other ingredients in it, since it remains brightly yellow and thick.

Bearnaise is a variation of hollandaise, which uses white wine vinegar to emulsify the egg yolks and butter, and flavors the resulting sauce with shallot, chervil, and tarragon. Like an aioli, it is not defined by the emulsion but by the flavoring agents. Adding terragon and capers, or tarragon and shallots, to a hollandaise (or other egg-yolk emulsion sauce) will produce a 'faux-bearnaise'.

The garlic defines the aioli, which is also the only one that can be made without egg yolk. The choice of butter rather than oil makes a hollandaise, and it is cooked as a sauce to set the yolks (which neither of the others require). The yolk and oil combination itself is the central concept of mayonnaise, while it can be flavored, it doesn't have to be.

The sauces are quite similar, though, and an emulsion of egg yolk with seasonings might be called by any name. The difference between a garlic mayo and a french-style aioli are likely to be pretty subtle. Likewise, a mayo flavored with lemon and pepper will be hard to distinguish from a hollandaise which uses oil - although hollandaise is usually cooked to set it, so the texture may be different. And a aioli with butter will be very similar to a hollandaise with garlic. Adding tarragon and shallot to any of them is likely to produce a bearnaise type sauce.

All three recipes are very loose and broadly defined, since they can be tweaked a lot depending on preferences - so it makes it hard to pin down other factors like ingredient ratios that might distinguish the recipes. The sauces that add flavorings or substitute ingredients will tend to be named one or the other based more on recipe origin, familiarity, and marketing rather than any clearly defined difference.

Yes, hollandaise is a warm mayo. The melted butter should denature the egg yolks a little bit, so even if it's made with olive oil, I'd expect there to still be some heat involved to classify it as 'hollandaise' and not simply 'mayonnaise'.

Aïoli that you see in restaurants is often just mayo with garlic in it, but the classic Spanish recipe is made with just garlic, olive oil, and salt -- no vinegar or other acid and no eggs, mustard or other emulsifiers. (It also requires a lot of effort -- as you have to crush up the garlic to a paste in a mortar & pestle, then work the olive oil in slowly 'til it turns into a thick sauce.)

Megha's answer covers contemporary usage among hobbyists. There is an alternative point of view: the standard classification schema for French sauces. It is still in use in posh restaurants today.

This schema absolutely precludes that "an emulsion of egg yolk with seasonings might be called by any name" and gives a separate name for dozens of possible variations. The simple "oil + yolk" variation does not exist there. "Mayonnaise" is the name for an emulsion of oil, yolk, acid and mustard. This is quite a deviation from the widespread usage today, and may be the reason why Ramsay chose to call his sauce a variant of hollandaise and not of mayonnaise.

The hollandaise is analogous to contemporary usage, an emulsion of yolks and butter (also with acid).

In the Careme classification, aioli is garlic + oil + yolk. The non-emulsified versions are not covered by his taxonomy.

Note that I am not insisting that one classification is more right or wrong than the other. But I have seen very heated discussions between people who were convinced that there is a single classification. So it is best to be aware of the whole picture.

Mayonnaise is an oil emulsion sauce made with beaten egg yolk and olive oil and lemon juice. Hollandaise is a butter emulsion sauce made with beaten egg yolk clarified butter (butter without its milk and water - just melt in a big pitcher. It is trickier to make than mayonnaise because if the butter "freezes" toward a solid or semi-solid point, the sauce will likely break, and if the egg yolk cooks from being too hot, the sauce will certainly break. Hollandaise is great on broccoli or warm artichokes. The trick learned from experience is to keep it from breaking. When I was cheffing at the Iron Men Inn in Iowa City, one of my prep jobs was to make a quart of Hollandaise/Bearnaise and keep it alive through the dinner service. The warm shelf above the stove was the right temperature, but most home kitchens don't have an equivalent warm space.

Bearnaise sauce is Hollandaise sauce with added minced shallots lightly reduced in butter and (hopefully fresh) tarragon. People who don't like tarragon may like the Texas Red variant below. Bearnaise is a good accompaniment for grilled steak and roasted beef or pork. It can be good on Barbecue, though it is definitely not traditional in North Carolina.

It is worth noting that an even more entertaining "Bearnaise-like" sauce may be made as follows (takes two days): Prepare Texas Chili - beef, ancho chilis (1-2 per pound), cumin and salt; brown the beef, which should be chuck, in medium chunks, pop the stems and seeds out of the anchos and toast them very lightly in another skillet until crisp, then break them into large pieces; stew beef and chilis all together with cumin and salt to taste and ABSOLUTELY NO OTHER ADDED VEGETAL INGREDIENTS of any kind. Pass beans or rice if you must. Reserve two or three cups of the broth before serving. Reduce these to a thick meat glaze jelly consistency. Use several tablespoons of this meat glaze jelly per finished cup -- and you should make at least a pint! -- of Hollandaise without much lemon, maybe just enough fresh organic lemon peel to brighten the flavor. This will be a knockout sauce, but ... you might add one or two drops of Lea & Perrins per cup, or a pea-sized kernel of anchovy paste for its meaty-flavor enhancing magic. N.B. -- do not use Pasilla chilis, they are too hot. Anchos are mild. Good luck and may the shade of Robert Courtine turn over in its grave. Although in truth -- he might approve.