Up until the late 60s, philosophy was dominated by a cluster of views very much influenced by logical empiricism:
The idea of that some kinds are more natural than others (say, water more natural than clean rooms) was frowned upon: all classifications should, rather, be thought of as reflecting human interests, and not as "carving Nature at its joints", as the phrase goes.
The idea that entities had some properties essentially or just accidentally was also frowned upon: maybe, the argument would go, Miguel Indurain is a bicycle rider only accidentally, but the winner of the Tour de France 1994 is necessarily a bicycle rider. But those two things are the same thing. So essentiality depends on perspective.
Relatedly, proper names were taken to be roughly synonymous with definite description: "Miguel Indurain" is synonymous with "The F", for some F.
All of these views were challenged by a number of immensely influential papers and books during the 70s. The two most important ones are Kripke (1980) and Putnam (1975).
In Putnam's paper it was claimed that some kinds (water, for example), are more natural than others in that they have an essence: a hidden property that explains its surface characteristics, and one that although it can be eventually uncovered by science, need not necessaritly be ever uncovered. In the case of water, the essence has been uncovered: it is Having molecular structure H-O-H. The kind of clean rooms is not natural at all precisely because it does not have such a hidden feature: it depends wholly on the interests of the users of the expression "clean room". Moreover, this appeals to essence, it was convincingly argued, was just uncovering what science had been tacitly relying on all along: scientists do not think, never thought, that everything that is, say, a yellow, malleable metal is gold: they had always assumed that there was some hidden feature about gold (not shared by iron pyrites, or fool's gold) that explains those properties (and there is: having atomic number 79).
Kripke argued (successfully, according to most) that there are facts about what is essential or contingent to the entities themselves, independently of the way in which they are described -- what have been called de re modal facts. For example, it is clear that Indurain could have not been a rider; and, once we keep in mind that we are talking of Indurain himself, we can use a definite description to express this fact: the winner of the Tour 1994 could have failed to be a bicycle rider.
Finally, this new realism about essences paved the way to a more apt theory of the semantics of proper names than descriptivism had ever been. Again, in Kripke (1980) we find the first version of the arguments, which remain compelling: Given the description that most competent speakers associate with a proper name such as "Miguel Indurain" -- one describing his succeses as a sportsman, most probably --, this description cannot be synonymous with the name. We have just seen that Miguel Indurain could have failed to be a sportsman at all, but if "Miguel Indurain" is synonymous with "the sportsman such that blah blah", then "Miguel Indurain could not have been the sportsman such that blah blah" would be a contradiction in terms. There are other arguments in the book I have just referenced; you might want to take a look. The more apt theory is one in which the reference of proper names is not secured via a definite description, but by causal contact between uses of the name and its bearer.
The result of this body of work was a true change of tide in contemporary philosophy: today, many accept the existence of de re modalities, and endorse one version or other of the causal theory of proper names -- and post-Kripke descriptivism bears little resemblance to pre-Kripke descriptivism.
This contemporary essentialism is, of course, a huge improvement above the Aristotelian variety. For one, it benefits from the advances in formal logic of the XXth century -- and one of the key developers of modal logic was Kripke himself --; it is also informed by contemporary science and contemporary linguistics, and, well, everything between Ancient Greece and now. If you have any more concrete interest on how contemporary essentialism differs from Aristotelian essentialism, let me know and I will try to elaborate (if I know anything about it).
Kripke, S. 1980. Naming and Necessity. Blackwell.
Putnam, Hillary. 1975. “The Meaning of ’Meaning’.” Minnesota Studies in the Philosophy of Science 7: 131–193.