According to Aristotle, a definition of X is a statement of X's essence. In contrast, contemporary philosophers often say that a definition of X as a set of individually necessary and jointly sufficient conditions for X. These two definitions of "definition" are not interchangeable.
Aristotle distinguishes between two kinds of necessary conditions. In (Latin) Aristotelian terminology, these two kinds of necessary conditions are called essence and proprium (plural: propria). A statement of X's essence captures what it is to be X. In other words, X's essence consists of the properties that make X X. In contrast, X's propria are the properties that necessarily follow from X's essence. For example, if the essence of a triangle is three-sided polygon, then one proprium of a triangle is having interior angles that add up to 180 degrees. Note that this proprium is a necessary condition for a triangle: if something doesn't have interior angles that add up to 180 degrees, then it isn't a triangle.[1] So, for Aristotle, essence and propria are two different kinds of necessary conditions. And I think that we have good reason to follow Aristotle in drawing this distinction.
I'm not making a metaphysical point here. The essence/propria distinction, as I am using it, doesn't depend on Aristotelian metaphysics with its "real essences". I'm perfectly happy to say that what we designate as a thing's essence is arbitrary from a metaphysical standpoint. We could say that a triangle's essence is three-sided polygon, or we could say that it's polygon whose interior angles add up to 180 degrees (since no other polygon has interior angles adding up to 180 degrees). There's no "real essence" inhering in triangles to guide our choice. (Or so I'm willing to grant for present purposes.)
Rather, my point is a conceptual one. When I apply a term, e.g. "triangle", I have a certain concept in mind, e.g. that of a three-sided polygon. Now, all sorts of properties may follow necessarily from my concept. If my concept is of a three-sided polygon, then it follows that any object matching that concept has interior angles adding up to 180 degrees. But those other properties aren't part of the concept just because they follow necessarily from it. Indeed, I may not even know that my concept entails those properties. If someone asked me to define a triangle, then I would try to articulate my concept of a triangle, not all the properties that follow from that concept.
Thus, even non-essentialist philosophers can make a sort of distinction between essence and propria. For there's a distinction between the properties contained in my concept of X and the properties that follow from that concept. In other words, there's a distinction between what I'm treating as the essence of X and the propria that follow from that essence.
There are some areas where the essence/propria distinction might actually contribute helpfully to disputes. One such area is the debate between legal positivists and natural law theorists. Oversimplifying greatly, we can describe the debate as follows: legal positivists think that law can be defined without any reference to morality; natural law theorists think that law can't be defined in a morally neutral way and that an immoral law (or a law that is immoral in certain ways) is not really a law. For example, legal positivists will often argue (following H.L.A. Hart) that a law is any rule that fits a given society's "rule of recognition", its standard for determining which rules are authoritative for it. Natural law theorists respond by pointing out that legal systems—and rules of recognition—are created for moral purposes: promoting peace and harmony, preventing violence, etc. Thus, every legal system that's actually fulfilling its purpose as a legal system is necessarily moral, at least in certain ways (it may be immoral or unjust in other ways, of course).
It seems to me that the legal positivists could respond by appealing to the essence/propria distinction. "What we're trying to describe," they might say, "is the essence of law. Law's essence—or at least the essence that we have in mind when we use the term 'law'—is a system of rules that fit a society's rule of recognition. Now, given that human beings care about morality, and given that they implement a rule of recognition in order to promote moral ends, it follows that every properly-functioning legal system is also moral, or at least not grossly immoral. This minimal level of morality is a proprium of legal systems. But it isn't their essence: their essence can be expressed in purely non-moral terms."
Of course, the natural law theorists could respond by including morality in the essence of law. But then they would be working with a different concept of law than the legal positivists—in which case the two camps would simply be talking past each other. There's no point in A and B arguing over the proper analysis of a concept unless they actually have the same concept in mind.
[1] Here I'm leaving aside complications that arise when we consider "non-Euclidean" geometry.
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