The Scholastic philosophers in the Middle Ages developed a theory of properties of terms in which different classifications of concepts feature prominently.
At the time of Kripke's lectures, the dominant theory of reference in analytic philosophy (associated with the theories of Gottlob Frege and Bertrand Russell) was that the meaning of sentences involving proper names could be given by substituting a contextually appropriate description for the name.
Russell,[3] for example, famously held that someone who had never met Otto von Bismarck might know of him as the first Chancellor of the German Empire, and if so, his statement that (say) "Bismarck was a ruthless politician" should be understood to mean "The first Chancellor of the German Empire was a ruthless politician" (which could in turn be analysed into a series of more basic statements according to the method Russell introduced in his theory of definite descriptions).
Kripke argued—against both the Russellian analysis and several attempted refinements of it—that such descriptions could not possibly mean the same thing as the name "Bismarck," on the grounds that proper names such as "Bismarck" always designate rigidly, whereas descriptions such as "the first Chancellor of the German Empire" do not.
The "could" here is important to note: rigid designation is a property of the way terms are used, not a property of the terms themselves, and some philosophers, following Keith Donnellan, have argued that a phrase such as "the first Chancellor of the German Empire" could be used rigidly, in sentences such as "the first Chancellor of the German Empire could have decided never to go into politics."
Proper names rigidly designate for reasons that differ from natural kinds terms.
This usage of 'Johnny Depp' for referring to some particular baby got passed on from person-to-person in a giant causal and historical chain of events.