What is the definition of a property? Like, for example, "that apple has the property of being red". I know that properties aren't the same thing as sets. But what are they, exactly?

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    Basic properties (and relations), like red, only have operational "definitions", that is procedures for determining if something has them (spectral frequency can be used for red). Composite properties can be defined in terms of basic properties and relations using logical connectives and quantifiers, e.g. red-or-green. – Conifold May 22 at 0:26

A very controversial issue in modern philosophy, as both your likening them to sets (or maybe Quinean classes) and @Conifold's arguably cryptic [though correct] comment. Though I am way out of my depth here, you may wish to peruse the following article, which will give you a fair working definition of what, traditionally, as elucidated by moderns, a primary property, or attribute [of an object] “is”, as opposed to it's secondary "qualities": https://www.academia.edu/6652802/The_distinction_between_Primary_Properties_and_Secondary_Qualities_in_Galileo_Galileis_Natural_Philosophy.

Here you will find the following:

[i]n this passage, Galileo in fact introduces a new definition of a body. A body in itself is a piece of matter with only primary properties. About twenty years later, Descartes would write in his Principia (1644) that a body is essentially an extended thing, a res extensa. So, a body is for the French philosopher a way or a mode of being extended. For Galileo, a body is not limited to this one unique property, but also includes several primary properties, such as size, motion, and shape [also number, location, and touching or not-touching another body]. A body is thus for Galileo, in 1623, a mode or a way of being a whole cluster of primary properties.

This new doctrine of properties-qualities is one of the elements of Galileo’s natural philosophy that make our common -sense pictures of objects in particular, and nature as a whole, illusory. Galileo argued that contrary to our intuition the sun does not move around the earth. He also argued in his new theory of motion, which he developed in the second part of his Dialogo (1632), that we don’t need a force to maintain the motion of a heavy body in a horizontal plane. On the contrary, we need an external force to stop such motion. And, importantly for this discussion, bodies in particular and corporeal substance in general do not have any sensible qualities such as odors, sounds or colors. On the contrary, our imagination (“imaginazione”) simply attributes these qualities to external bodies, so that they can be stripped off again by means of our imagination (see Galilei, 1890-1909: VI, 348). Primary properties, by way of contrast, cannot be stripped off. For example, a ball would not be the same body if we stripped off its spherical form. Galileo would argue that it can be geometrically demonstrated with certainty that the ball is spherical and not triangular or cubic.

As well as the following quote taken from Edwin A. Burtt's The Metaphysical Foundations of Modern Physical Science. A Historical and Critical Essay:

This form of the primary secondary distinction in Galileo is worth a moment’s pause, for its effects in modern thought have been of incalculable importance. It is a fundamental step toward that banishing of man from the great world of nature and his treatment as an effect of what happens in the latter, which has been a pretty constant feature of the philosophy of modern science, a procedure enormously simplifying the field of science, but bringing in its train the big metaphysical and especially epistemological problems of modern philosophy.

Please do not ask me to elaborate this answer further. For I can not. And fear that I would be forced, embarrassingly, to refer you to David Hume or John Locke; before Kant irredeemably muddled the waters.

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