A single paper everyone should read? Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues. 
Do you have such an example?
Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.
And as usual let's do one per post and vote freely, vote a lot.
 A: Perhaps not really a paper, but i think a "must-read" is A Mathematician's Lament  by Paul Lockhart.
A: Andre Weil's "Two lectures on number theory. past and present." L'Enseignement Mathematique. Revue Internationale. fie Serie. 20: 87-110. 1974
available here http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001:1974:20::43&subp=hires
Great historical perspective on number theory up to the early 1970's. Easy to read too!
[I should remark that despite the article's great virtues, Weil is (apparently) unfair to Hardy and that many topics in number theory are left untouched.]
A: William Thurston's On Proof and Progress in Mathematics is a wonderful read, enlightening many aspects of the practice of mathematics.
A: An Elementary Theory of the Category of Sets
http://tac.mta.ca/tac/reprints/articles/11/tr11abs.html
I always had a problem with ZFC because it makes too many arbitrary choices: why do we choose this countable set to be the natural numbers and not this other one?  Why do we choose Kuratowski ordered pairs instead of some other version?  This paper turned me on to the idea that all of mathematics could be done in a "nice" way, where things are only determined up to unique isomorphism by the properties you want them to satisfy.  It was also my first exposure to category theory, and so holds a special place in my heart. 
A: Cannon's beautiful and accessible paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the original impetuses for geometric group theory.  It inspired many people (including me) to become interested in infinite discrete groups.  It is available here:
https://doi.org/10.1007/BF00146825
A: I would argue for Shannon's "A Mathematical Theory of Communication". Its wonderfully written, started an entire field of research (or two), and struck a very nice balance between abstraction and transparency in the mathematics.  The ideas first introduced in that paper are powerful tools even today!
A: 2N Noncollinear Points Determine at Least 2N Directions, by Peter Ungar.
This is a beautiful short paper that proves the result in the title.
A general remark: If you have to choose a single paper (or a single paper of a mathematician selected in other answers), I would recommend more strongy to choose original papers of important basic results rather than large survey papers or  "meta" paper about mathematics. (This is also closer to the original intention of the question.)  
A: I highly recommend this lucid, little book (with the length of a paper):
Mathematics: A very short introduction, by Fields Medalist Timothy Gowers
A: I had recommended to me from several prominent faculty the paper:

The Yang-Mills Equations over Riemann Surfaces
Author(s): M. F. Atiyah and R. Bott
Source: Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615
Published by: The Royal Society
Stable URL: https://www.jstor.org/stable/37156

One professor called it "the basis for truly 21st century mathematics." It is also reportedly accessible by beginning graduate students with some exposure to differential geometry and suitable for independent study or as a reading course. It is a 93 page paper and develops a lot of fundamental constructions and ideas from scratch. Here is
Martin Guest's review on MathSciNet.
A: Missed Opportunities, Freeman Dyson
A: "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" by Jonathan Shewchuk at UC Berkeley
A: I would recommend Gowers' The Two Cultures of Mathematics. It talks about the two types of mathematicians, the "theory builders" and the "problem solvers." 
A: Another suggestion: A beginner's guide to forcing by Tim Chow.
It really explains the continuum hypothesis, in a very accessible and captivating way. People often talk about the continuum hypothesis, but it's nice to know what's going on for real.
A: Carl's Pomerance "A tale of two sieves", available at
http://www.ams.org/notices/199612/pomerance.pdf
It makes a quick introduction to subexponential factoring algorithms via their development from Fermat's Algorithm and then compares the Quadratic Sieve with Her Majesty the (General) Number Field Sieve, in a thorough, appealing and very understanable manner.
A: Toen's course on stacks.  I don't know if this counts as a paper, but courses 2,3, and 4 introduce a really interesting approach to geometry using the functor of points approach that I've not seen before.
A: PDE as a Unified Subject   by Sergiu Klainerman.
An essay on partial differential equations written by a leading expert in the field, I strongly recommend to anyone who aspires to know more on the subject as well as to those who are not interested strictly in PDE's, but would like to get a grasp of interactions between Mathematics and Physics. There are also many interesting references.
A: I think "What is good mathematics?" by Terry Tao is a great paper because it argues that we do not need to all be pursuing the same ideal of good mathematics (and indeed, people should pursue disjoint ideals), and it provides an interesting case study of a nice result, Szemerédi's theorem.
A: "On the Number of Primes Less Than a Given Magnitude", B. Riemann.
A: One paper that I want to share with any of my colleagues, although it is not in my field, is Doyle and Conway, Division by Three, math/0605779v1.
To emphasize why this paper is so great, let me quote the entirety of the conclusion (saving you the trouble of reading the rest of the paper):
 What’s wrong with the axiom of choice?
Part of our aversion to using the axiom of choice stems from our view that it 
is probably not ‘true’. A theorem of Cohen shows that the axiom of choice is 
independent of the other axioms of ZF, which means that neither it nor its 
negation can be proved from the other axioms, providing that these axioms 
are consistent. Thus as far as the rest of the standard axioms are concerned, 
there is no way to decide whether the axiom of choice is true or false. This 
leads us to think that we had better reject the axiom of choice on account 
of Murphy’s Law that ‘if anything can go wrong, it will’. This is really no 
more than a personal hunch about the world of sets. We simply don’t believe 
that there is a function that assigns to each non-empty set of real numbers 
one of its elements. While you can describe a selection function that will 
work for ﬁnite sets, closed sets, open sets, analytic sets, and so on, Cohen’s 
result implies that there is no hope of describing a deﬁnite choice function 
that will work for ‘all’ non-empty sets of real numbers, at least as long as 
you remain within the world of standard Zermelo-Fraenkel set theory. And if 
you can’t describe such a function, or even prove that it exists without using 
some relative of the axiom of choice, what makes you so sure there is such a 
thing?
Not that we believe there really are any such things as inﬁnite sets, or that 
the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. 
Indeed, we’re somewhat doubtful whether large natural numbers (like 805000, 
or even 2200) exist in any very real sense, and we’re secretly hoping that 
Nelson will succeed in his program for proving that the usual axioms of 
arithmetic—and hence also of set theory—are inconsistent. (See [E. Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.]) 
All the more reason, then, for us to stick with methods which, because of their 
concrete, combinatorial nature, are likely to survive the possible collapse of 
set theory as we know it today.
A: That's easy just off the top of my head, Illya: Nets And Filters In Topology by the late Robert G. Bartle; appearing in the 1955 Volume 62 of American Mathematical Monthly. I remember having a friend in the Stanford mathematics honor society who'd published papers by age 20,but had never heard of either nets or filters. I recommended it to him right on the spot.
A: If you are a geometer I would say  it is worth to read the paper of Gromov, called
"Spaces and Questions",  this paper is not about one single result, it rather gives a point of view on geometry, which seems very inspiring, at least to me, here is the reference:
https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SpacesandQuestions.pdf
A: I would have to go with the "Five Worlds" paper by Impagliazzo. It is a beautiful overview of how many complexity/cryptographic results relate to each other and what they "mean" for the real world (as of 1995, at least). It is a great way to web all those buzz words from class and coffee discussions into a cohesive unit.
-Yan
A: Birds and Frogs by Freeman Dyson, which explains nicely that the world of mathematics is both , broad and deep.
A: Imre Lakatos "Proofs and Refutations". Great book about origin of mathematical reasoning and rise of formal theories.
A: On the theorem of Pythagoras by E.W. Dijkstra.
(Did you know that in every plane triangle $\operatorname{sgn}(\alpha + \beta - \gamma) = \operatorname{sgn}(a^2 +  b^2 - c^2)$, a "theorem, say, 4 times as rich [as the original]"?)
A: In recent years Manin has put out several philosophical writings on mathematics, physics, and other related topics:
Truth as value and duty: lessons of mathematics 
Mathematical knowledge: internal, social and cultural aspects
The notion of dimension in geometry and algebra
Georg Cantor and his heritage
Von Zahlen und Figuren
There's also a book, Mathematics as Metaphor, that collects even more of Manin's philosophical material.
These are all very nice reads and I would recommend them to almost anyone, mathematician/physicist or not.
A: "On Computable Numbers, with an Application to the Entscheidungsproblem", Alan Turing, 1936.  A great mind and a great paper.
A: Stallings's How Not To Prove the Poincare Conjecture (cached at Citeseer) is the funniest paper I've ever read. 
A: Advice to a Young Mathematician in the Princeton Companion to Mathematics 
A: Proofs from the Book! (Ok it's a book rather than a paper, but just pick any chapter.) Every line is amazing.
A: If you ever - as in my case - quoted a textbook to your students claiming that pointwise convergence of Fourier series for piecewise continuous functions is difficult and subtle, you'll feel stupid after reading Paul Chernoff's two-page paper "Pointwise Convergence of Fourier Series."
I can't find a free online copy of it, but you should be able to read it here with university access: JSTOR (Actually, you can see the first page for free, which already proves the main result.)
(Or get it from the library:
The American Mathematical Monthly, Vol. 87, No. 5 (May, 1980), pp. 399-400.)
EDIT: There is a free copy here:
https://web.archive.org/web/20100730044517/http://math.berkeley.edu/~strain/118.S10/chernoff.pointwise.convergence.of.fourier.series.pdf
A: Not technically a paper but a lecture (in pdf form) full of pretty pictures and cool ideas:
The Mysteries of Counting: Euler Characteristic versus Homotopy Cardinality
by John Baez.

We all know what it means for a set to have 6 elements, but what sort of thing has -1 elements, or 5/2? Believe it or not, these questions have nice answers. The Euler characteristic of a space is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive real values. These concepts shed new light on basic mathematics. For example, the space of finite sets turns out to have homotopy cardinality e, and this explains the key properties of the exponential function. Euler characteristic and homotopy cardinality share many properties, but it's hard to tell if they are the same, because there are very few spaces for which both are well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series---and the two then agree! The challenge of unifying them remains open.

A: Two additional papers in combinatorics (That I managed to find on line) each having a beautiful and simple result.
On the Shannon Capacity of a Graph by Laszlo Lovasz
The Upper Bound Conjecture and Cohen Macaulay Rings by Richard Stanley
A: Paul Halmos How to Write Mathematics
A: I like Musical Actions of Dihedral Groups pretty much. It gives a nice view of harmony (the art of using chords in music), considering the set of chords as the dihedral group of order 24 (12 major + 12 minor). 
Unfortunately, this is useful only for people into music and maths. I would also like to share it with my musician friends, but most of them will probably run away at the sight of the first mathematical term...
Please don't vote down if you're not a musician ;).
A: "On the Electrodynamics of Moving Bodies", Albert Einstein
A: Two notes on notation by Knuth. This paper discusses "Iverson" notation, which is of use to almost all mathematicians, and good notation for Stirling numbers.
A: The Unreasonable Effectiveness of Mathematics in the Natural Sciences
by Eugene Wigner
Although Wigner is physicist, I consider this article about mathematical physics very important both for physicists and mathematicians. It's a wonderful feeling to realize to what extent our world can be mathematical.

The miracle of the appropriateness of
the language of mathematics for the
formulation of the laws of physics is
a wonderful gift which we neither
understand nor deserve. We should be
grateful for it and hope that it will
remain valid in future research and
that it will extend, for better or for
worse, to our pleasure, even though
perhaps also to our bafflement, to
wide branches of learning.
E.Wigner

A: I am surprised to see that so many people suggest meta-mathematical articles, which try to explain how one should do good mathematics in one or the other form. Personally, I usually find it a waste of time to read these, and there a few statements to which I agree so wholeheartedly as the one of Borel: 
"I feel that what mathematics needs least are pundits who issue prescriptions or guidelines for presumably less enlightened mortals."
The mere idea that you can learn how to do mathematics (or in fact anything useful) from reading a HowTo seems extremely weird to me. I would rather read any classical math article, and there are plenty of them. The subject does not really matter, you can learn good mathematical thinking from each of them, and in my opinion much easier than from any of the above guideline articles. Just to be constructive, take for example (in alphabetical order)


*

*Atiyah&Bott, The Yang-Mills equations over Riemann surfaces.

*Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts.

*Furstenberg, A Poisson formula for semi-simple Lie groups.

*Gromov,Groups of polynomial growth and expanding maps.

*Tate, Fourier analysis in number fields and Hecke's zeta-functions.


I am not suggesting that any mathematician should read all of them, but any one of them will do. In fact, the actual content of these papers does not matter so much. It is rather, that they give an insight how a new idea is born. So, if you want to give birth to new ideas yourself, look at them, not at some guideline.
A: One paper that I've read a few times and always loved was Who Can Name the Bigger Number? (also available in Spanish and French, for those who prefer to read in those).  It discusses how our concept of "big numbers" has evolved over time, and talks about Turing machines and the "busy beaver" numbers, which represent a non-computable function.
A: "Rigor and Proof in Mathematics: A Historical Perspective" by Israel Kleiner. Mathematics Magazine December 1991, 64:291-314.
This paper gives a very nice overview of how the understanding of rigor in mathematics has evolved from the early ages to the 20th century.
https://www.jstor.org/stable/2690647
