I was among those who discussed Zariski's paper on simple points on the MO thread. Here is the link.
One landmark paper is that of Deligne and Mumford on moduli spaces of curves. (It appeared in Publications IHES, and would be easy to track down.) It will need more than Hartshorne Chapters I, II, and III, but could well provide an incentive to learn that little bit more.
As I've mentioned in other threads on this topic, I think that Mumford's book Lectures on curves on algebraic surfaces is fantastic. (It is longer than a paper, but it is devoted to the proof of a single result. Along the way, it develops a lot of fantastic material and intuitions.)
Serre's GAGA paper is another classic.
Finally (until I think of more must-adds!) there is the paper of Clemens and Griffiths, The intermediate Jacobian of the cubic threefold. Since this may seem a little specialized, let me exlain why I think it deserves classic status:
a smooth cubic curve in the plane is not rational (it has genus one); a smooth cubic surface in space is rational — it is $\mathbb P^2$ blown up at six points. A smooth cubic threefold in $\mathbb P^4$ was classically known to be unirational, but (before this paper) it was not known whether or not it was rational; this paper shows that it is not rational. Questions of rationality are fundamental in algebraic geometry, and this paper is a fundamental contribution; it also marks Griffiths's introduction of Hodge-theoretic ideas (the modern point of view on periods of integrals as studied by Abel and Picard, and later Lefschetz) as key tools in the study of concrete geometric questions. Note that the problem of rationality of cubic fourfolds remains open.
Okay; some more classics that came to mind while I was writing: Atiyah's paper on Vector bundles over an elliptic curve (one should first read Grothendieck's paper on vector bundles on $\mathbb P^1$), and (to give a more recent example) the paper of Graber–Harris–Starr, proving that the total space of a family of rationally connected varieties over a rationally connected base is rationally connected.
More: Variations on a theorem of Abel (I think this is the right title), by Griffiths. If you want to understand what the Abel–Jacobi theorem (and hence what Hodge theory and much else in modern algebraic geometry) might really be about, in concrete geometric terms, this is a paper you must read.
Deligne's note Théorie de Hodge I and his paper Théorie de Hodge II are also fantastic. (There is also part III, but it is more technical, since it deals with singular varieties.) There is a precursor, something like On a criterion for the degeneration of spectral sequences (but in French). These papers, like those of Griffiths that I've mentioned, mark the introduction of Hodge theory into modern algebraic geometry as a fundamental tool. Deligne's style is very different to Griffiths's; it is harder to see the concrete meaning of what he is doing than in Griffiths's work. But they are both masters, introducing ideas that are as fundamental and influential as any that I can think of in geometry.