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.