Suggested Reading for CogSci Grad Students
The papers and books below are among those that I think every graduate student in cognitive science could benefit from reading, regardless of their area of specialization. Needless to say, there is just as much additional 'required reading' within any given area of specialization...  
The current version of this document is: 9/7/14.

Helpful suggestions and additions for this list were provided by: Chris Chabris, Zenon Pylyshyn, Jonathan Weinberg.  
Please forward suggested additions to  
General Scientific Methodology
Platt, J. R. (1964). Strong inference. Science, 146, 347-353.
[Very highly recommended!]  
Kuhn, T. S. (1970). The structure of scientific revolutions, 2nd Edition. Chicago, IL: University of Chicago Press.  
Cohen, J. (1992). Fuzzy methodology. Psychological Bulletin, 112, 409-410.  
McGuire, W. J. (1997). Creative hypothesis generating in psychology: Some useful heuristics. Annual Review of Psychology, 48, 1-30.
[I'm unsure how ultimately useful advice of this sort is, but this is the best example I've seen of some heuristics for generating research ideas.]  
Koenderink, J. J. (2002). The head and the hands [Guest Editorial]. Perception, 31 517-520.
[Some helpful thoughts about being a experimenter vs. a theoretician.]  
Ioannidis, J. (2005). Why most published research findings are false. PLoS Medicine, 2(8), e124.
[Now that's a provocative title, eh? For a popular treatment, see Jonah Lehrer's 12/13/10 New Yorker article, called "The truth wears off: Is there something wrong with the scientific method?"]  
Statistics & Psychological Methodology
Sedlmeier, P., & Gigerenzer, G. (1989). Do studies of statistical power have an effect on the power of studies? Psychological Bulletin, 105, 309-316.  
Gigerenzer, G., et al. (1989). The inference experts. Chapter 3 of The empire of chance: How probability changed science and everyday life. Cambridge: Cambridge University Press.
[This book is a history of statistics, with a focus on how statistics became integrated into various disciplines and into culture more generally. Chapter 3 in particular focuses on the development of the Fisherian and Pearsonian views of statistics, a crude combination of which infest today's psychological practice. Very interesting, and well worth reading!]  
Cohen, J. (1994). The earth is round (p < .05). American Psychologist, 49, 997-1003.  
Dar, R., Serlin, R., & Omer, H. (1994). Misuse of statistical tests in three decades of psychotherapy research. Journal of consulting and clinical psychology, 62, 75-82.  
Roberts, S., & Pashler, H. (2000). How persuasive is a good fit? A comment on theory testing. Psychological Review, 107, 358-367.
[Nuanced discussion of the utility and pitfalls of fitting a model to a data set. Too many such models aren't really useful, for the reasons discussed here.]  
Mellers, B., Hertwig, R., & Kahneman, D. (2001). Do frequency representations eliminate conjunction effects? An exercise in adversarial collaboration. Psychological Science, 12(4), 269-275.
[For this purpose, ignore the stuff on frequency representations and focus on the method of adversarial collaboration!]  
Dixon, P. (2003). The p-value fallacy and how to avoid it. Canadian Journal of Experimental Psychology, 57, 189-202.
[A nice primer on why you shouldn't rely too much on ".05".]  
Gigerenzer, G. (2004). Mindless statistics. Journal of Socio-Economics, 33, 587-606.
[The best summary I've read about what's wrong with statistical practice today! This presents a summary of some of the same ideas from the Gigerenzer et al. (1989) chapter cited above, but in an updated and focused form. Entertainingly and aggressively written.]  
Durgin, F., Baird, J., Greenburg, M., Russell, R., Shaughnessy, K., & Waymouth, S. (2009). Who is being deceived? The experimental demands of wearing a backpack. Psychonomic Bulletin & Review, 16, 964-969.
[This is the best example I have read in recent years of how and why it is important to avoid task demands in your experiments! Utterly devastating.]  
Henrich, J., Heine, S., & Norenzayan, A. (2010). The weirdest people in the world? Behavioral and Brain Sciences, 33, 61-135.
[Some thoughts on how a constrained subject population might impede the search for truth.]  
Simmons, J., Nelson, L., & Simonsohn, U. (2011). False-positive psychology: Undisclosed flexibility in data collection and analysis allows presenting anything as significant. Psychological Science, 22, 1359-1366.
[A contemporary must-read paper, this summarizes several key aspects of methodology and analysis that are necessary in order to be either an informed consumer or producer of research!]  
Theoretical Underpinnings of Cognitive Science
Turing, A. (1950). Computing machinery and intelligence. Mind, 59, 433-460.
[Widely reprinted, e.g. in Hofstadter & Dennett's The Mind's I. Ignore the bit on ESP.]  
Chomsky, N. (1959). A review of B. F. Skinner's 'Verbal behavior'. Language, 35, 26-58.
[This book review ended up changing an entire field. Widely reprinted, e.g. in Block's Readings in the Philosophy of Psychology.]  
Putnam, H. (1979). Reductionism and the nature of psychology. Cognition, 2, 131-146.
[An early influential statement of why reductionism won't work for psychology. Widely reprinted, e.g. in Haugeland's Mind Design.]  
Gould, S. J., & Lewontin, R. C. (1979). The spandrels of San Marco and the Panglossian program: A critique of the adaptationist programme. Proceedings of the Royal Society of London, 205, 281-288.
[An antidote to careless adaptationism. An infamous paper, widely reprinted.]  
Searle, J. (1980). Minds, brains, and programs. Behavioral and Brain Sciences, 3(3), 417-457.
[Skim the commentaries too, and remember that arguments needn't be right to be important and instructive! This one is also widely reprinted, e.g. in Hofstadter & Dennett's The Mind's I.]  
Fodor, J. A. (1983). Modularity of mind. Cambridge, MA: MIT Press.  
Pylyshyn, Z. W. (1984). Computation and cognition: Toward a foundation for cognitive science. Cambridge, MA: MIT Press.
[Tough going at parts, but full of subtle, extremely important ideas.]  
Churchland, P. M. (1988). Matter and Consciousness, Revised Edition. Cambridge, MA: MIT Press.
[A great summary of the various -isms: behaviorism, functionalism, dualism, eliminative materialism, etc.]  
Dennett, D. C. (1995). Darwin's dangerous idea. New York: Simon & Schuster.
[Or any of several other introductions to evolutionary theory, oriented to the cognitive sciences.]  
Pinker, S. (1997). How the mind works. New York: Norton.
[A very well-written summary of the state of the art circa the late 90s across much of cognitive science, written from an 'evolutionary psychology' perspective, for the lay-reader.]  
The History of Psychology
James, W. (1890/1950). The principles of psychology. New York: Dover.
[The single most important historical book you can read. Very highly recommended.]  
Mandler, G. (2011). A history of modern experimental psychology: From James and Wundt to cognitive science. Cambridge, MA: MIT Press.
[It's always nice to know at least something about the history of your field!]  
The Sociology of Science
Ramon y Cajal, S. (1897/2004). Advice for a young investigator. Translation reprinted by: Cambridge, MA: MIT Press.
[Extremely useful thoughts on getting started in science, from more than 100 years ago.]  
Campbell, D. T. (1969). Ethnocentrism of disciplines and the fish-scale model of omnicience. In M. Sherif & C. W. Sherif (Eds.), Interdisciplinary relationships in the social sciences (pp. 328-348). Xenia, OH: Aldine.
[While many of the specific ideas in this chapter strike me as wrong or misguided, this is an excellent framework for thinking about how to find an appropriate research topic and initial career path -- and also an excellent stimulant for thinking about the organization of our field.]  
Nisbett, R. (1978). A guide for reviewers: Editorial hardball in the 70s. American Psychologist, May, 519-520.
[Humorous and dead-on characterization of the sociology of reviewing papers in our field. Still relevant, alas.]  
Nisbett, R. (1990). The anticreativity letters: Advice from a senior tempter to a junior tempter. American Psychologist, September, 1078-1082.
[Thoroughly excellent and characteristically well written advice on how (not) to best position yourself for scientific success.]  
Thompson, K. S. (1994). Scientific publishing: An embarrassment of riches. American Scientist, 82, 508-511.  
Greene, M. T. (1997). What cannot be said in science. Nature, 388, 619-620.  
Rozin, P. (2006). Domain denigration and process preference in academic psychology. Perspectives on Psychological Science, 1, 365-376.
[An important meditation on how typical foci of research in psychology depart from what most people care about.]