What is your favorite isomorphism? The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't even listed in its index. The Wikipedia article on isomorphisms only gives two concrete examples.
There are many surprising, significant, classic isomorphisms. I'll refrain from giving examples. What are your favorites?
As usual, please limit yourself to one isomorphism per answer.
(Related: your favorite surprising connections in mathematics. But this question is looking for more concrete examples, particularly those that illustrate the power of the idea.)
 A: One of my current favourites can be found on Peter Cameron's blog.
Take a countable model $(M,E)$ of enough (axioms of) set theory. 
Symmetrize the relation $E$ to obtain a graph.  This graph is the random graph
(Rado's homogeneous universal countable graph).
A: The De Rham Isomorphism.
A: The isomorphism between $SL_2(\mathbb{C})$ and the universal covering of the special Lorentz group $SO^+(1, 3)$ is definitely nifty in my opinion. ("Coincidences" between Lie groups are another good source of examples.) 
A: The whole subject of non-commutative geometry arises from extending to non-commutative algebras the isomorphism that exists between commutative $\mathrm{C}^*$-algebras and locally compact Hausdorff topological spaces.
A: Here is an example that Mel Hochster used to explain the notion of isomorphism to a group of talented high school students.  I was one of the course assistants rather than one of the students, but I'm sure the insight was at least as valuable for me as for them.
Consider the following game.  I'll write down the numbers 1 through 9 on a sheet of paper, and you and I will take turns selecting numbers from the list (crossing off each number once it has been selected).  The winner is the first person to have chosen exactly three numbers which add up to 15.  For example if I selected 9, 6, 2 and you selected 3, 8, 1, 4 then you would win because 3 + 8 + 4 = 15.
The interesting thing is that this game is isomorphic to tic-tac-toe.  I don't know what I precisely mean by that, but I can explain why it is true.  Simply consider a 3 x 3 magic square:
4 9 2
3 5 7
8 1 6
The rows, columns, and diagonals all add up to 15, and moreover every way of writing 15 as the sum of three numbers from 1 to 9 is represented.  When you choose a number, draw an X over it; when I choose a number, circle it.  Tic-tac-toe!
A: I like the isomorphism between a finite abelian group and its "Cartier" dual (not the bidual!) precisely because it's non-canonical. But I don't think it makes a good example for explaining isomorphism to non-mathematicians.
A: A calculator made using wire and logic gates with electrons flowing through it should be considered isomorphic to a calculator made using tubes and physical gates with water flowing through it as long as the underlying structure (the schematics for each calculator) is the same.
A: I should say I'm fond of the Thom isomorphism, but I still find the contents rather mysterious.
A: English and French are isomorphic.  
Stronger. They are both trivial.
See this paper by Mestre, Schoof, Washington, and Zagier for a short proof.  
A: The set of positive reals under multiplication is isomorphic to the set of reals under addition, which is the isomorphism underlying the operation of a slide rule.  This is the only isomorphism I can think of important enough that its explicit (approximate) values used to be published in 1000-page books.  The positive reals under multiplication is also a standard pedagogical example of an interesting one-dimensional abstract real vector space, where there is some content to verifying the axioms.  (The other standard example is the reals with addition given by $x+y-1$ and multiplication by scalar $a$ given by $ax + 1 - a$.) 
A: The most striking example of an isomorphism I remember seeing as an undergraduate was when John Conway visited and gave his famous talk on rational tangles.  Being able to unknot a seemingly hopelessly tangled pair of skipping ropes by manipulating rational numbers was an amazingly concrete demonstration of what it meant for structures to be isomorphic.
For those who don't know what I'm talking about, I think there's a video online somewhere.
A: The Cantor pairing
function is
the function $p(a,b)= (a+b)(a+b+1)/2 + b$, a polynomial
bijection between the pairs of natural numbers and
individual numbers. Thus, it is a bijection or isomorphism
of the sets $\mathbb{N}\times\mathbb{N}$ and $\mathbb{N}$.
Using such a function, one may easily deduce that the set
of rational numbers is countable, and more generally, that
a countable union of countable sets is countable.
A: I nominate the Chinese Remainder Theorem, in the form of an isomorphism of a ring of residues with a cartesian product ring. This isn't "profound" mathematics, but simply unpacking it (with construction of the underlying idempotents) should convince students that algebraic structure has "content". I recall a conversation about the analogue for polynomials in one variable over a finite field, in which my side was really stating that if you understand the original CRT in the correct way, this is no sweat at all.
A: I'm a fan of the isomorphism between $PSL_2 (F_7)$ and $GL_3 (F_2)$ (two nice descriptions of the simple group of order 168).  Years ago, Richard Guy asked me if I knew an explicit map, and I didn't.  But recently one was given in the Math Monthly:
MR2572107
Brown, Ezra; Loehr, Nicholas
Why is ${\rm PSL}(2,7)\cong{\rm GL}(3,2)$?
Amer. Math. Monthly 116 (2009), no. 8, 727--732.
The paper is also available from Brown's website:
http://www.math.vt.edu/people/brown/doc.html
A: The elliptic modular function
j(τ) = q-1 + 744 +196884q + ...  (q=e2πiτ) 
This is an isomorphism from elliptic curves (such as C/(1,τ))  to the  complex plane.
A: The set of 7-tuples of binary trees is isomorphic to the set of binary trees. For the correct definition of "isomorphic" this is a surprising non-trivial result.
A: How about the Fourier transform as an isomorphism between the Hilbert space $L^2$ of quadratically integrable complex-valued functions on the unit interval and the Hilbert space $\ell^2$ of sequences of complex numbers the sum of the squares of whose norms is finite?
