Most math papers have few figures, if any, although sometimes a wellchosen figure can be a tremendous help in understanding mathematical concepts. Does anyone have any examples of notable uses of figures in mathematical writing and/or texts that make great use of figures/diagrams/illustrations?

4$\begingroup$ I find it interesting that, while the answers contain many links to fabulous diagrams, no one has actually incorporated any of these diagrams into his/her answer. $\endgroup$ – Gerry Myerson Jun 17 '15 at 4:55
Mumford's picture of Spec(Z[x]) comes to mind.
You can read a discussion of it here.
I am a huge fan of the drawings of Anatoly Fomenko (for some of his drawings, see here; for a description of his odd historical theories, see his wikipedia pages here and here).
In particular, his book "Algorithmic and Computer Methods for ThreeManifolds" with Matveev (which really has nothing to do with computers) is IMHO one of the best intro books on 3manifolds available largely because of its drawings. The mathscinet review of the Russian version is worth reading; see
MR1162113 (93f:57002) Matveev, S. V.; Fomenko, A. T. {\cyr Algoritmicheskie i kompʹyuternye metody v trekhmernoĭ topologii}. (Russian) [Algorithmic and computer methods in threedimensional topology] Moskov. Gos. Univ., Moscow, 1991. 303 pp. ISBN: 5211017439
EDIT : To give an idea of how amazingly cool the above book is, I have posted two pages of it here and here. It's a crying shame that it is not better known...

3$\begingroup$ I love his artwork. Do you have any idea if anyone makes prints of his work? $\endgroup$ – Beren Sanders May 16 '10 at 23:05
A paper that used figures to, in my view, revolutionize the understanding of an area of mathematics is: R. Penrose. Applications of negative dimensional tensors. In D.J.A. Welsh, editor, Combinatorial mathematics and its applications, pages 221–244. Mathematical Institute, Oxford, London, New York, Academic Press, 1971.
Turning a sphere inside out
The book by G. K. Francis entitled "A topological picturebook" is beautiful. Have a look at this snapshot, starting with page 16.
The book explains how to draw and visualize pictures of low dimensional famous topological spaces: the dunce hat, a tetrahedral hyperbolic manifold, the Withney bottle, the Hopf fibration, and so on.
When I was an undergraduate I read the book Intuitive Topology by Prasolov. It's a wonderful illustrated guide to lowdimensional topology (mostly knots/links and surfaces). If I recall correctly, almost all of the "proofs" are by pictures.
Edward Tufte's books are quite beautiful, though they do not focus so much on mathematical figures/diagrams per se. However, via Tufte, I did come across this version of Euclid's Elements by Oliver Byrne, which presents the propositions and proofs of the Elements using colored diagrams and symbols. I'm not sure whether Byrne's edition is clearer or better to learn from than the original Euclid, but it sure is pleasing to look at.

$\begingroup$ Sergei Slyusarev has started a project to TeX Byre's book: github.com/jemmybutton/byrneeuclid/releases/tag/0.4%2B0.1 $\endgroup$ – Ben McKay Jun 28 '17 at 15:53
Reid's Undergraduate Commutative Algebra has a lot of great figures, most notably the frontispiece depicting the statement "let $A$ be a ring and $M$ an $A$module" geometrically.

1$\begingroup$ It's a great image, but a somewhat provocative placement, making the frontispiece of “Undergraduate Commutative Algebra” a shibboleth which will delight the initiate but baffle a newcomer to Algebraic Geometry. But then, Miles Reid doesn't tend to shy away from being provocative... $\endgroup$ – Peter LeFanu Lumsdaine Sep 22 '10 at 14:32
Geometry in Figures "written" by Arseniy Akopyan consist only of Figures. From Preface:
This book is a collection of theorems (or rather facts) of classical Euclidean geometry formulated in figures. The figures were drawn in such a way that the corresponding statements can be understood without any additional text.
Here's one of my favorite diagrams, based on a paper by Lawrence Leemis:
Probability distribution relationships
Also, here are some diagrams due to Robert Bartle relating various modes of convergence:
There's a chapter in Littlewood's "Mathematician's Miscellany" entitled The Zoo, which has many charming examples of pictoral proofs inspired by various animals.
Knots, links, braids, and 3manifolds, by Prasolov and Sossinsky (you can look at it on Google books) essentially has a picture on every page. They are very pretty.
Indra's pearls by Mumford, Series, and Wright has some breathtaking pictures. There are also cartoons by Gonick, which improves any book. Here's the copy at Google books.
Edit: A topological picturebook, by Francis is wonderfully illustrated. There are directions for reproducing the figures, as well as their mathematical meaning. Chapter eight of the book deals with the figure eight knot. :)
see my starting effort to help see 3manifolds at http://commons.wikimedia.org/wiki/Category:3manifolds
The following is a wondeful candidate: Jos Leys, Lorenz and Modular Flows: A Visual Introduction, Feature Column of the AMS web site, November 2006.
The figures in John Stillwell's books are always superbly drawn, and really enhance the exposition.
BarNatan's first paper on Khovanov homology included a great figure (visible immediately if you follow that link) that summarised the entire construction. A great improvement over the previous epic.
Arnol'd was famous for his pictures (that poor $\ $ stretched $\ $ cat...), but the Award for Best Pictures must surely go to "Geometry and the Imagination" by D. Hilbert and S. CohnVossen.
H.S.M. Coxeter's books include Regular Polytopes. This book deals with the classification of regular polytopes. In this book there are Coxeter diagrams which are closely related to Dynkin diagrams. In his works are many diagrams,figures and illustrations. They influenced M.C Escher. Many of Escher's works reflect his ideas. In 1996 Coxeter published a paper on one of these "Circle Limit III." For more information see here and here.
The standard diagram for representing a category, showing objects, arrows/morphisms and the composition operation:
(The required identity arrows are elided here, which I find reasonable but which could easily be added without disturbing the diagram too much if their absence bothers you.)