The ADE type Dynkin diagrams seem to come up in seemingly different areas of math. Two places they come up are:
(1) Classification of simply laced complex simple lie algebras.
(2) Finite subgroups of $Sl_2 (\mathbb{C})$
Are there any other objects that they classify?
They classify quivers for which the path algebra is of finite representation type., acording to a famous theorem of P. Gabriel.
They classify the so called "simple singularities" of differential maps, that is, of those types of singularities which involve no parameters. See V. I. Arnolʹd, S. M. Guseĭn-Zade, A. N. Varchenko, Singularities of differentiable maps, Vol. 2, Chap. 15, sect. 1.
As Mariano said, the ADE Dynkin diagrams classify quivers of finite representation type. But wait, there's more. If you add one more vertex to a Dynkin diagram (in a particular way, not an arbitrary one), you get an extended Dynkin diagram (aka a Euclidian diagram). The extended ADE diagrams classify quivers of tame representation type. This is related to the fact that the extended ADE diagrams give you a positive semi-definite Tits form, while the ordinary ADE diagrams give you a positive definite Tits form.
They parametrize the finite dimensional preprojective algebras.
They classify the cluster algebras of finite type (that is, with a finite number of clusters). See S. Fomin, A. Zelevinski, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63--121.
They classify the (germs of) isolated rational singular points of two dimensional complex analytic spaces. See A. E. Durfee, Fifteen characterizations of rational double points and simple critical points. Enseign. Math. (2) 25 (1979), no. 1-2, 131--163.
They classify principal graphs of $II_1$-subfactors with index less than $4$. The principal graph can be $A_n$, $D_{2n}$, $E_6$, or $E_8$, but $D_{odd}$ and $E_7$ do not occur.
This paper http://arxiv.org/abs/hep-th/0006151 (CFT, BCFT, ADE and all that, by Jean-Bernard Zuber) provides the following list of mathematical objects that fall in an ADE classification:
simple simply-laced Lie algebras, i.e. with roots of equal length;
finite reflection groups of cristallographic and of simply-laced type;
finite subgroups of SO(3) or of SU (2), (or the associated platonic solids);
Kleinian singularities;
“simple” singularities, i.e. with no modulus;
finite type quivers;
symmetric matrices with eigenvalues between −2 and +2;
algebraic solutions to the hypergeometric equation;
subfactors of finite index;
Some others, presumably not listed in the answers given, include:
Smooth Freund-Rubin backgrounds of eleven-dimensional supergravity (http://arxiv.org/abs/0909.0163 - Half-BPS quotients in M-theory: ADE with a twist, by Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia and Elena Méndez-Escobar).
Non-smooth Freund-Rubin backgrounds of eleven-dimensional supergravity (http://arxiv.org/abs/1007.4761 - Half-BPS M2-brane orbifolds, by Paul de Medeiros and José Figueroa-O'Farrill)
Articles explaining the appearance of ADE classification from within string theory are listed at http://ncatlab.org/nlab/show/ADE+classification
David Corfield already indicated John Baez's blog "This Week's Finds in Mathematical Physics", Week 230, where you can find a link to van Hoboken thesis http://www.jorisvanhoboken.nl/?p=10 (Platonic Solids, Binary Polyhedral Groups, Kleinian Sigularities and Lie Algebras of Type A,D,E).
They classify certain types of rational conformal field theories, as in this recent review paper.
This article gives a nice overview on the "ADE-problem". It is written in the late 1970's, so it does not cover more recent appearences. Where is their appearance is most surprising?
Extended Dynkin diagrams appear naturally in Kodaira classification of singular fibers of an elliptic surface.
Let $G$ be a connected graph with the property that all eigenvalues of $G$ lie in $[-2,2]$ (such a $G$ is called cyclotomic). Then $G$ is either one of $\tilde{E}_6,\tilde{E}_7,\tilde{E}_8$, an $\tilde{A}_n$ for $n\ge 2$, a $\tilde{D}_n$ for $n\ge4$, or an induced subgraph of one of these. In other words, the ADE graphs classify the maximal cyclotomic graphs.
They classify prehomogeneous actions (=actions with finitely many orbits) of $GL(n,\mathbb C)$ on non-trivial exterior representations.
Those are $\mathbb C^n$, $\Lambda^2\mathbb C^n$, and $\Lambda^3\mathbb C^n$ for $n=6,7,8$
up to the symmetry $\Lambda^k\mathbb C^n\leftrightarrow\Lambda^{n-k}\mathbb C^n$.
The coincidence $A_3=D_3$ corresponds to $\Lambda^1\mathbb C^3\leftrightarrow\Lambda^2\mathbb C^3$.
You might also take a look at Slodowy, P. (1983), Platonic Solids, Kleinian Singularities and Lie Groups,Lecture Notes in Mathematics, No. 1008, pp. 102- 138,
Arnold's Trinities paper "Polymathematics : is mathematics a single science or a set of arts?", easily found on the net
and
Chapoton's own trinities page (also found on the net).
The last two focus on the "E" part of ADE, but give long lists of intriguing parallels.
I suggest to take a look on a very nice Givental's paper (MR1138519 (92k:58031)):
"Reflection groups in singularity theory."
Here is the review (by V.D. Sedykh): The simple singularities of functions are classified by the Coxeter groups $(A,D,E$-classification). This classification arises in other problems, too (the classifications of the simple Lie algebras, of the finite quaternions groups and so on). The author gives a detailed survey of these results. He also considers the problems connected with the classification of the quasihomogeneous unimodular singularities of functions (the classification of the degenerations of elliptic curves, the theory of automorphic functions and so on) in this paper.
The ADE root systems classify(/are classified by) the generalized quadrangles that have three points on each line. See Line Graphs, Root Systems and Elliptic Geometry by Cameron, Goethals, Seidel and Shult; Chapter 12 of Algebraic Graph Theory by Godil and Rolye; and these notes by A. E. Brouwer.
