What you are talking about are called universals.
To know what something is (here, that some concrete thing you've encountered is a cat) is to know its form. The form of a thing is that which causes it to be what it is. By "cause", I do not mean efficient cause, but what is called the formal cause. The form is the principle by which a thing is what it is, by which it is the kind of thing it is. Here, you've identified the thing as a Cat. In the case of a ball made of bronze, the Sphericity is the form that causes the ball to be what it is (a ball; the bronze is the material cause); if you hammer the ball into a cube, you have changed what the thing in question is.
Of course, the judgement you are making in your mind is something like "this thing is a Cat", where "this thing" is the subject, and "Cat" is the predicate. But how does this predicate come be to? It's not "out there", in the world. You didn't see it. The form Cat isn't what you've encountered, but some concrete instance. The predicate exists in your intellect, but how did it get there? The key is abstraction. This is the essential function of the intellect. Abstraction is the intellectual power by which the form (or essence, nature) of a thing can be separated from a thing, as made available in the senses, and sustained apart from any particular objects that instantiate it. This is a very unique property of intellects, as any other occurrence of form in the world is in the concrete things that instantiate it themselves. You never encounter forms in the real. You never see Cat, only this case or that cat. Or, if you like, this triangle, or that triangle, but never Triangularity as such.
The features of the instance in question are called accidents. Now, the form I mentioned above is known as the substantial form (as integral wholes, like individual cats, in the world are what philosophers call substances). But accidental form is all that cannot be instantiated as such. Red fur, blue eyes, long hair, even having four paws or a heart - these are accidents. Why? Because while you can instantiate cats as such, you cannot instantiate "red fur", or just "red", as such, only as part of some substance (someone might object that you can have red fur on your sofa that exists apart from a cat; this is true, but it is no longer fur as such, but something that was fur, just like a severed hand is no longer a hand, because it no longer functions as a hand). Those accidents that follow necessarily from the nature of the substantial form are called properties. For some time, philosophy has become rather loose and sloppy with this word, using it to mean any kind of accidental feature, but traditionally, a property was something proper to something by virtue of the kind of thing that it is. So while red fur is not a property, just an attribute, having a heart or four paws are arguably properties, because it is essential to what a cat is.
Of course, just having a bag of attributes or properties doesn't help us much. They must inhere in some whole, and the most important thing you can know about something is what it is. Is it the what of a thing that provides the ground by which we can make sense of all the characteristics of that thing. So it makes sense that the what is what is intellectually most prominent or primary. "Red" or even "red fur" by themselves don't help you understand what you've encountered.
However, you might object: "But wait, I don't always know what a thing is! But I can see its features!" Yes. When you encounter something for the first time, you may not have a good understanding of the nature of the thing. This is true. But you do have some understanding, however minimal, which is developed the more you observe and interact and study it. When I say that it makes sense that the essence should come to mind first, I do not mean to say it is the first thing you encounter. Note what I wrote earlier, that we abstract from particulars as given to use in the senses. Sense perception is prior to abstraction, or else you would not have anything to abstract from. And what we sense are the accidents, as essence is not something we sense as such, but something we abstract from what is sensed. So we come to know the essence through a things accidents. The whole is known before its parts or features as such, which are abstracted. Thus, there is never a moment when you judge "I see red fur" full stop. Rather, the senses, encounter a whole (the cat), and the intellect is where the judgement you spoke of, e.g., "oh, look, a cat" and "this cat has red fur", is occurring. The world is only made known, not merely sensed, by the intellect. There is never a moment when you merely sense and then judge, as judgement is intellectual and thus already past the point when abstraction has already occurred.
