Math has a really serious PR problem, and it's one of my long-term goals to change this. As a small step, I'm collecting here a sampling of toy examples and analogies that can be explained to an interested non-mathematician. Please feel free to use these! And please feel free to suggest more such stories, as well.
- K-theory: We study spaces by how we can "glue little toothpicks" to them in a continuous way. (For the cognoscenti, "toothpick" = "vector space".) For example, the circle has two ways of gluing toothpicks: one gives a cylinder, the other gives a Möbius band. On the other hand, only the first extends continuously over the disk. Thus, K-theory detects the difference between the circle and the disk.
- Schemes: Any space has a "ring of functions". A ring is just a set where you can add, subtract, and multiply; in the ring of functions on a space, these operations are all taken pointwise. On the other hand, there are many interesting rings which are clearly not the ring of functions on any space. Grothendieck was not happy with this state of affairs, and so gave rigorous foundations for a beautiful and productive way to think of any ring as "the ring of functions on a space". This allows for one to study algebraic problems (such as those arising in number theory, the study of the ring Z of integers) via geometric methods and intuition.
- Algebraic K-theory: Play the "K-theory" game with "schemes". When you apply this to Spec(Z), you get something very closely related to many deep theorems and conjectures in number theory.
- Differential topology: The Gauss--Bonnet theorem typifies the relationship between local geometry and global topology. A good example is a deflated soccer ball: the creased part "makes up for" the negatively-curved part.
- Topological spaces: Configuration spaces are a great example of topological spaces that appear in real life, whose topology matters. Consider the possible configurations of a robotic arm with many joints; or the possible positions and bearings that a Mars rover can reach from some initial conditions (with only certain moves that it can perform, and operating on non-flat terrain); or the possible configurations a bunch of forklifts moving along a factory floor (modeled by configurations of points on a graph).
- Arithmetic: The passage from each number system to the next is motivated by the solution of some new class of problems. Starting from the natural numbers, we invent the integers when studying subtraction, the rationals when studying division, the reals when studying limits, and the complex numbers when studying polynomials. This leads to the fundamental theorem of algebra, which admits many different proofs (algebraic, analytic, topological) that each illustrate a useful technique from a different area of mathematics (Galois theory, maximum principles, fundamental groups).
- E∞-spaces: An E∞-space is what you get when you only ask your multiplication to be "commutative up to coherent homotopy". An E2-space is probably easier to describe, since it's easier to visualize points colliding in a plane. Additionally, this makes clear the possibility of "homotopical ambiguity": there's a circle's worth of multiplications, so all multiplications are equivalent and yet the space of them isn't contractible.
- Vector fields on spheres: The hairy ball theorem can be phrased as saying that at any given point in time, there must be some point on earth with no wind (assuming wind defines a continuous vector field, of course). To go further, consider "two different types of wind", say hot and cold (or red and blue, or whatever); you can ask whether it's possible for these two winds to always be blowing in different directions (though of course this is already impossible on the 2-sphere). An n-dimensional space is called parallelizable if there's a way of having n different types of wind that are always pointing in linearly independent directions. An amazing fact (proved using "K-theory") is that the n-sphere is parallelizable if and only if n is 0, 1, 3, or 7. The cases 0 and 1 can be easily visualized.
- Convergence of power series: A power series with real coefficients has radius of convergence determined by its complex poles. This is shocking.
- Spectral sequences: These are like books, where each page is a grid with certain dots filled in. To find out what's on the next page, you have to solve a bunch of problems, figuring out how the dots interact. When you reach the end, you find out a secret fact (the most secret fact of all being "the music of the spheres"!). The deeper the secret is, the harder the problems are to solve.