Sauces that split and need to be shaken to come back together present an unstable emulsion. And those that stay uniform no matter the time are examples of a stable emulsion.
To put it in simple terms, emulsion is a mixture of usually non-mixable products, such as oil & water.
If you just mix oil & water, they could mix for a limited amount of time, depending on how vigorously you mix them, but eventually would split nevertheless.
BUT! If you introduce some kind of an emulsifier to such emulsion, like lecithin (from egg yolks) in the examples of mayonnaise & sauce hollandaise, or dairy casein in butter and chocolate, then such emulsions won’t split.
Et voila! Now you know what mayo, butter & chocolate have in common.