Behind any and all forms of essentialism, I think, there lies a basic intuition, which I will call minimal classificatory essentialism (MCE): there are kinds of things; and the kind of thing a thing is, is relevant to what it can do and undergo. A corollary of this is that distinction among kinds is, at least in principle, more than merely conventional. Or, in other words, it is at least in principle possible for us to classify things correctly, because there is a right and wrong about classification.
To deny minimal classificatory essentialism is irrational. Such a denial entails that it is impossible to know what things are, or why they do what they do, or why they undergo what they undergo -- because it is a denial that there is any 'what' or 'why' to know. So, clearly, anyone who is rational must be an essentialist at least so far; and, a fortiori, any scientific work must presuppose such an essentialism. If MCE were false, no classification and no identification of properties whatsoever would tell us anything about the world. Thus we all have a strong transcendental argument for accepting at least MCE, and we always have.
Naturally, however, it is possible to have a stronger form of essentialism; and we can identify two types of stronger essentialism: shared-nature essentialism and type essentialism. A key difference: shared-nature essentialism allows that the kind can be defined. Type essentialism (in its pure, exclusive form, at least) denies this, saying instead that kind is determined by relation to type.
The difference can be made clearer by examining a typical type essentialist argument. The following argument is T. H. Huxley's useful summary of William Whewell's biological type essentialism:
It is said, in short, that a natural-history class is not capable of being defined--that the class Rosaceae, for instance, or the class of Fishes, is not accurately and absolutely definable, inasmuch as its members will present exceptions to every possible definition; and that the members of the class are united together only by the circumstance that they are all more like some imaginary average rose or average fish, than they resemble anything else.
Now, the language here is slightly tendentious, but only slightly. By 'imaginary' we need to understand 'idealized' and by 'average' we need to understand 'normal'. That is, the type essentialist holds that the kind can only be determined by idealizing and abstracting from variations, and recognizing that the actual particulars cluster around these idealized reference points.
Huxley in his criticism of Whewell, for which see the above link, makes statements that technically commit him to being a shared-nature essentialist, since the view that rigorous definition of kinds is possible is a distinguishing mark of shared-nature essentialism. I'm not sure if this is consistent with Huxley's overall view. Certainly Darwin's own view does not appear to involve shared-nature essentialism at all. But it should be noted that it is possible for someone to be both. The reason for this is that Type and Shared Nature do not constrain kinds in the same way. Individuals of a kind are constrained to approximate conformity with a type; they are constrained by a shared nature. Thus it is possible to be both. However, we must understand that there is no general position 'Shared Nature Essentialism' or 'Type Essentialism'; if you are a shared nature essentialist, you are so for a given domain, and so it is with type essentialism as well. It is possible, for instance, to accept shared nature essentialism for chemical kinds, but deny it for biological species. Whewell, for instance, seems to have denied shared nature essentialism only for biology, because he thought definitions were, even in principle, impossible for biology, but not for other things. [*] We cannot give a definition of rose, because there is too much variation; but we can identify an idealized rose (having such-and-such features) around which all roses cluster, exhibiting greater or lesser conformity to type. (And note that we can subdivide; thus, if there were a sub-clustering of roses, that would tell us that there is a type there, too.) Types are, as it were, the attractors of classification.
[* In retrospect this is too strong. Whewell doesn't deny shared nature essentialism; he merely denies that it is useful for scientific practice, i.e., for all practical purposes biologists must resort to types.]
Of course, it would be entirely possible for us to hold that, strictly speaking, the position here that deals with essences (at least directly) is shared nature essentialism; and we could speak of this as essentialism (in the proper sense) and use another term for type essentialism. But in fact, as far as I can tell, no one does this when talking about essentialism. It's easy enough to find recognition that people thinking in terms of shared natures are thinking in a different way that people thinking in terms of types, however related the two may be; but the distinction doesn't seem to be made much when talking about essentialism. It is fairly clear, for instance, that type essentialisms are usually swept into the category 'essentialism' when people talk about the history of biology, because that's the only way sense can be made of things like (for instance) talk about Darwin versus essentialism. We should, however, be cautious of assuming that something true of shared nature essentialism would always be true of type essentialism, and vice versa.
In my next post, I think, I'll try to sketch out the sorts of essentialisms one finds in biology, or, at least, the most important ones that were around in the nineteenth-century (there were a lot).
UPDATE, 15 March: Fixed a few of the most obvious typos and misstatements.