Saul Kripke, answered from the texts and cited to the page.
A proper name is a rigid designator: it designates the same object in every possible world in which that object exists, and never designates anything else.1 Names are not synonymous with definite descriptions, and a name's reference is not determined at an arbitrary world via the satisfaction of a descriptive condition at that world.2 The reference is fixed at the actual world — typically by an ostensive baptism, or occasionally by a stipulation that the name shall designate whatever satisfies a description — and once fixed, the name designates that individual at every world where the individual exists.3
The descriptive content a speaker happens to associate with the name plays no role in determining who the name refers to.4
What do I mean by 'rigid designator'? I mean a term that designates the same object in all possible worlds. To get rid of one confusion which certainly is not mine, I do not use "might have designated a different object" to refer to the fact that language might have been used differently.1. Identity and Necessity, pp. 9–10
More important for present purposes, though, than accepting this NAMING AND NECESSITY 57 example as an instance of the contingent a priori, is its illustration of the distinction between 'definitions' which fix a reference and those which give a synonym. In the case of names one might make this distinction too. Suppose the reference of a name is given by a description or a cluster of descriptions.Naming and Necessity, pp. 65–66
It seems plausible to suppose that, in some cases, the reference of a name is indeed fixed via a description in the same way that the metric system was fixed. When the mythical agent first saw Hesperus, he may well have fixed his reference by saying, 'I shall use "Hesperus" as a name of the heavenly body appearing in yonder position in the sky.Naming and Necessity, p. 66
David Kaplan has suggested that a demonstrative "that" can be used, in English, to make any definite description rigid. "That bastard—the man who killed Smith, whoever he may be—is surely insane!" The subject term rigidly designates Smith's murderer, but it is still attributive in Donnellan's sense.5. Speaker’s Reference and Semantic Reference, p. 106
Defended in Naming and Necessity through the modal argument (the meaning of "Aristotle" cannot be "the teacher of Alexander" because Aristotle might not have taught Alexander, yet "Aristotle might not have taught Alexander" is true), the epistemic argument (the meaning cannot be a description if the descriptive identification is the kind of thing one could discover empirically), and the semantic argument (the speaker may attach a wrong description and still refer — the Goedel-Schmidt case). Held without modification across the subsequent corpus. Soames-style friendly extensions (partially descriptive names; semantic content of names in attitude contexts) refine the picture without overturning the anti-descriptivist core.