Describe a topic in one sentence. When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good high-level view (sometimes!) What I'm looking for are good one-sentence descriptions about a topic that deliver the (or one of the) main punchlines for that topic.
For example, when I look back at linear algebra, the punchline I take away is "Any nice function you can come up with is linear." After all, multilinear functions, symmetric functions, and alternating functions are essentially just linear functions on a different vector space. Another big punchline is "Avoid bases whenever possible."
What other punchlines can you deliver for various topics/fields?
 A: Configuration space integrals: Don't take limits- compactify!
Dror Bar-Natan explained this punchline to me when I was just starting grad school.
A: "set theory is the study of well-foundedness" - A.R.D Mathias
A: Terry Tao, in a post on Google Buzz, has given an overview of mathematics in the form of multiple "punch-lines" of the requested variety.
Here are three examples from Tao's post:


*

*Algebra is the mathematics of the "equals" sign, of identity, and of the "main term"; analysis is the mathematics of the "less than" sign, of magnitude, and of the "error term".


*Algebra prizes structure, symmetry, and exact formulae; analysis prizes smoothness, stability, and estimates.


*Most of geometry would not be classified as either algebra or analysis, but simply as geometry.

Definitely Tao's aphorisms are thought-provoking and inspiring ... but are they useful ?  Don't ask me!  :)

Partly inspired by Tao's essay, here is a one-sentence definition of quantum mechanics (as optimized for systems engineers)  …


*

*Quantum mechanics is the algebraic geometry of $n$-particle Hamiltonian flows and Lindbladian compressions as pulled-back onto the natural $r$-indexed stratification of $r$'th secant varieties of $n$-factor Segre varieties whose $r\to\infty$ limit is … $n$-particle Hilbert space.


… and it turns out to be very useful (and great fun) to rewrite standard quantum physics texts like Charles Slichter's Principles of Magnetic Resonance based upon this one sentence definition.
Joseph Landsberg's recent Bull. AMS review "Geometry and the complexity of matrix multiplication" (2008), which has been praised in multiple MathOverflow posts, provides an overview of the broad utility—despite their unwieldy name—of  stratifications of secant varieties of Segre varieties (which extends far beyond quantum physics).
A: About Sobolev spaces, Meyers and Serrin theorem H = W 
A: Geometric representation theory: keep translating the problem until you run into Hard Lefschetz, then you are done.
A: I'll offer two punchlines for Galois Theory. 


*

*There's a one-to-one, order-reversing correspondence between intermediate fields of a finite, normal, separable extension $K$ of $F$, and subgroups of the group of automorphisms of $K$ fixing $F$. 

*A polynomial is solvable in radicals if and only if the Galois group of its splitting field is a solvable group. 
A: Homological algebra - In an abelian category, the difference between what you wish was true and what IS true is measured by a homology group.
A: Algebraic Topology: Geometry is hard, and Algebra is easy so...
(I am sure this applies to many other fields, and certainly algebra is hard.)
A: Etale cohomology - you can apply fixed-point theorems from algebraic topology to Galois actions on varieties.
A: Functional analysis: Everything you know from linear algebra is true, under the right conditions; otherwise it's false.
A: Complex Analysis: Holomorphic functions are just rotations and dilations up to the first order.
Hold on...
Calculus: Differentiation is approximation by a linear map.
A: Lie groups: Think locally, act globally. ;)
A: Analysis: Allez en avant, et la foi vous viendra (D'Alembert, to a student who had difficulty in believing the calculus of infinitely small. Translation: go on, and faith will be bestowed on you :)
A: Four-Dimensional Smooth Manifolds: Whitney's trick gone wrong.
A: Complex Analysis:  Taylor series behave the way you want them to in real analysis.
A: Real Analysis: Get your hypotheses right, or suffer the counter-examples!
Measure Theory: "Every [measurable] set is nearly a finite union of intervals; every [measurable] function is nearly continuous; every convergent sequence of [measurable] functions is nearly uniformly convergent." -- J.E. Littlewood
A: One punchline in algebraic geometry is that all commutative rings are actually the ring of functions on some space.
A: Analytic combinatorics: generating functions are awesome.
("generating functions are awesome" is actually the title of a talk I gave a couple weeks ago.)
A: Operator theory: all separable infinite-dimensional Hilbert spaces are isomorphic, but they aren't all the same and moving your problem between them works wonders.
A: *

*Generating functions are the 19th Century analog of addressable memory.
A: Nonlinear optimization: Newton's method beats everything else (when it works); when it doesn't, do something that looks like Newton's method.
A: Quantization: First quantization is a mystery, second quantization is a functor (Edward Nelson)
A: Representation theory of compact groups: The representation theory is the same as for finite groups, only that there might be infinitely many isomorphism classes of irreducible representations. 
(That's the Peter Weyl Theorem!)
Perhaps it would be  a much better question, to interpret a well known theorem in one sentence!
A: Numerical analysis: The purpose of computing is insight, not numbers. — Richard Hamming (1962)
A: One of my favorites:
"Algebraic topology is the "art" of Not doing the integral"
A: Homotopy theory is an attempt to do homological algebra in non-abelian categories. 
A: Algebraic geometry: CommRing behaves a lot like Setop.
A: The bonniest mot I can ever recall — from some graduate algebra course:


*

*"Free" is just another word for nothing to do on the left.
A: Linear algebra: everything can be explained by a linear system.
A: Navier-Stokes Equations: Energy estimates and more energy estimates. 
*I suppose this goes for most non-linear PDEs
A: Another favorite of mine …


*

*Redundancy is the essence of information.
A: Set theory without choice: You have no choice, but to wonder...
Forcing: If it doesn't not fit, force it.
Large cardinals: "If you want more you have to assume more." (Dana Scott)
A: Additive combinatorics: Any two attempts to define what it means for a finite set to be `additively structured' will be approximately equivalent.
A: Number Theory : Arithmetic properties (such as number of rational solutions) of geometric objects (such as elliptic curves) are often reflected in analytical functions (such as L-functions) associated to those objects i.e. geometry reveals its arithmetic analytically.
A: Representation theory of Lie groups: there is a whole world between $\mathrm{Sym}^n V$ and $\wedge^n V$. (Okay, this is an oversimplication - I am talking about the representations of $\mathrm{GL}\left(V\right)$ here, but this is the fundament of all other classical groups.)
Constructive logic: if you can't compute it, shut up about it. (At least some forms of constructive logic. Brouwer seemed to have a different opinion iirc.)
Homological algebra: How badly do modules fail to behave like vector spaces?
Gröbner basis theory: polynomials in $n$ variables can be divided with rest (at least if you have some $O\left(N^{N^{N^{N}}}\right)$ of time)
Finite group classification: what works for Lie groups will surely be even simpler for finite groups, right? ;)
Algebraic group theory: In order to differentiate a function on a Lie group, we just have to consider the group over $\mathbb R\left[\varepsilon\right]$ for an infinitesimal $\varepsilon$ ($\varepsilon^2=0$).
Semisimple algebras: The representations of a sufficiently nice algebra mirror a structure of the algebra itself, namely how it breaks into smaller algebras.
$n$-category theory: all the obvious isomorphisms, homotopies, congruences you have always been silently sweeping under the rug are coming back to have their revenge.
Modern algebraic geometry (schemes instead of varieties): let's have the beauty of geometry without its perversions.
How many of these did I get totally wrong?
A: Logic teaches us that (untrained) intuition is often wrong, but that when it's right, it's for the wrong reason.
A: Analytic Number Theory: log log log log log...
Did I see that quote in Havil's book Gamma?
A: Renormalization in quantum field theory: "just because something is infinite doesn't mean it is zero". (Explanation: this was said in about 1950 when regularization/renormalization was discovered as a way of getting sensible non-zero values for formally infinite expressions.)
A: Noncommutative Ring Theory: If it is not modules, then it is idempotents.
A: Topological Vector Spaces: You can make an infinite dimensional space have every nice property of finite dimensional spaces- but not all of them at once.
A: Geometric group theory: the large-scale geometry of a group is invariant under quasi-isometry.
A: Algebraic geometry is the study of the intrinsic properties of any mathematical object which can be locally described by polynomial equations.
Or
Algebraic geometry is not about solving systems of polynomial equations, rather it's about studying the intrinsic properties thereof.
A: Dirichlet forms: a symmetric Markov process is a self-adjoint operator is a closed symmetric form is a Markovian semigroup.
(I've left out a lot of hypotheses, but the essence is that all these are in correspondence, and the properties of any one appear in the others.)
A: I think this belongs on this list too:
The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.    – James Newman
A: Morse Theory: opus dynamicum maxime.
A: Harmonic analysis: The integral operator with the kernel (blank space to fill in) is bounded from (blank space to fill in) to (blank space to fill in).
(communicated by Mark Rudelson) 
A: Probability/Statistical mechanics:
Take a probabilistic model (possibly complicated, involving huge state space, describing a complex system) and rescale it suitably, such that in the limit a simpler "macroscopic" object emerges;
if the latter is still random it's a central limit theorem, if it's deterministic it's a law of large numbers, if you look at fluctuations from the latter it's large deviations; if it is largely independent on the details of the starting probabilsitc model, you have a universality phenomenon (and are happy because when modelling your real system you were forced to add some assumptions just for mathematical comfort); if it changes qualitatively when playing with a parameter of the original model you have a phase transition and want to know the critical values of the parameter.
A: QFT — every expression converges after a Wick rotation.
