First of all, my view on essentialism (or essences) can be given without recourse to a particular theory of mind, but depending on the mind theory you adopt, I will have different things to say about exactly ‘what’ the essence of things finally are.
Having said that, my view on essence, in plane words, is that the essence of a thing is what that thing ultimately is in reality. I don’t think essences belong to classes of things, but only to particular things (and usually only for a particular moment in time). So, for example, there is no essence of man as such, but each man, in a moment of time, has his own essence. He is, after all, something ultimately!
Like I said, though, the question of what this something is can differ from one theory of mind to another. To the niave realists, it is exactly what you perceive it to be. The essence of a tree, for example, is just the tree that you see before you - leaves, branches, trunk, roots, etc. - and exactly as you see it. To an idealist or a Kantian metaphysicist, the essence of things is something beyond perception, but just as real as you or I. The tree, for example, may not be exactly the green leaves or the rugged trunk or the moist roots, etc., but some imperceptible, perhaps inconceivable, thing outside one’s perceptual range. That thing, however, is nonetheless represented by the tree as perceived, and if you were to see the tree somehow perish (say in a fire) and its ashes be scattered about such that there is no more tree, the same would have to be said of the essential tree that exists beyond the realm of perception (so there is no permanence in this realm either).
I suppose you could count the platonic theory of mind whereby the conception of things (as in the conception of ‘mankind’) is a sort of ‘seeing’ into a realm of ‘forms’, and on this theory, my fundamental rendition of ‘essences’ falls apart, but like I said, I ain’t no platonist.