From Strings to Mirrors
To tell you where mirror symmetry came from, I have to tell you about string theory. And to do that, I have to tell you why you should care about string theory in the first place. That story starts with an old question: what is the smallest piece of the universe? ...
Introduction
Scientists often use mathematical ideas to make discoveries. The area of research mathematics known as mirror symmetry does the reverse: it uses ideas from theoretical physics to create mathematical discoveries, linking apparently unconnected areas of pure mathematics.
String theory
To tell you where mirror symmetry came from, I have to tell you about string theory. And to do that, I have to tell you why you should care about string theory in the first place. That story starts with an old question: what is the smallest piece of the universe?
Here's a rapid summary of a couple of thousand years of answers to this question in the West. The ancient Greeks theorized that there were four elements (earth, air, fire, water), which combined in different ways to create the different types of matter that we see around us. Later, alchemists discovered that recombination was not so easy: though many chemicals can be mixed to create other chemicals, there was no way to mix other substances and create gold. Eventually, scientists (now calling themselves chemists) decided that gold was itself an element, that is, a collection of indivisible components of matter called gold atoms. Continued scientific experimentation prompted longer and longer lists of elements. By arranging these elements in a specific way, Dmitri Mendeleev produced a periodic table that captured common properties of the elements and suggested new, yettobe discovered ones. Why were there so many different elements? Because (scientists deduced) each atom was composed of smaller pieces: protons, neutrons, and electrons. Different combinations of these subatomic particles produced the chemical properties that we ascribe to different elements.
This story is tidy and satisfying. But there are still some weird things about it: for example, protons and neutrons are huge compared to electrons. Also, experiments around the beginning of the twentieth century suggested that we shouldn't just be looking for components of matter. The electromagnetic energy that makes up light has its own fundamental component, called a photon. The fact that light sometimes acts like a particle, the photon, and sometimes like a wave is one of the many weird things about quantum physics. (The word "quantum" is related to "quantity"—the idea that light might be something we can count!)
Lots and lots of work by lots and lots of physicists trying to understand matter and energy particles, over the course of the twentieth century, produced the "Standard Model." Protons and neutrons are made up of even smaller components, called quarks. Quarks are held together by the strong force, whose particle is a gluon. The weak force, which holds atoms together, has its own force particles. The full Standard Model includes seventeen different fundamental particles.
There are two theoretical issues with the Standard Model. One is essentially aesthetic: it's really complicated. Based on their experience with the periodic table, scientists suspect that there should be some underlying principle or structure relating the different types of particles. The second issue is more pressing: there's no gravity particle. Every other force in the universe can be described by one of the "force carrier" particles in the Standard Model.
Why is gravity different? The best description we have of gravity is Einstein's theory of general relativity, which says gravitational effects come from curvature in the fabric of spacetime. This is an excellent way to describe the behaviors of huge objects, such as stars and galaxies, over large distances. But at small distance scales (like atoms) or high energies (such as those seen in a particle accelerator or in the early universe), this description breaks down.
People have certainly tried to create a quantum theory of gravity. This would involve a force carrier particle called a graviton. But the theory of quantum physics and the theory of general relativity don't play well together. The problem is the different ways they treat energy. Quantum physics says that when you have enough energy in a system, forcecarrier particles can be created. (The timing of their appearance is random, but it's easy to predict what will happen on average, just as we know that when we flip a coin over and over, we'll get tails about half the time.) General relativity says that the shape of spacetime itself contains energy. So why aren't we detecting random bursts of energy from outer space, as gravitons are created and destroyed?
String theory is one possible answer to this question. String theory says that the smallest things in the universe are not point particles. They extend in one dimension, like minuscule loops or squiggles— thence the name string. Strings with different amounts of energy correspond to the particles with different properties that we can detect in a lab.
The simplicity of this setup is compelling. Even better, it solves the infinite energy problem. Interactions that would occur at a particular moment in spacetime, in the point particle model, are smoothed out over a wider area of spacetime if we model those interactions with strings. But string theory does pose some conceptual complications. To explain them, let's look at the underlying mathematical ideas more carefully.
In general relativity, we think of space and time together as a multidimensional geometric object, fourdimensional spacetime. Abstractly, the evolution of a single particle in time is a curve in spacetime that we call its worldline. If we start with a string instead of a point particle, over time it will trace out something abstractly twodimensional, like a piece of paper or a floppy cylinder. We call this the worldsheet. One can imagine embedding that worldsheet into higherdimensional spacetime. From there, we have a standard procedure to create a quantum theory, called quantization.
If we work with fourdimensional spacetime, we run into a problem at this point. In general relativity, the difference between time and the other, spatial dimensions is encoded by a negative sign. Combine that negative sign with the standard quantization procedure, and you end up predicting quantum states—potential states of our universe, in this model—whose probability of occurring is the square root of a negative number. That's unphysical, which is a nice way of saying "completely ridiculous."
Since every spatial dimension gives us a positive sign, we can potentially cancel out the negatives and remove the unphysical states if we allow our spacetime to have more than four dimensions. If we're trying to build a physical theory that is physically realistic, in the sense of having both bosonic and fermionic states (things like photons and things like electrons), it turns out that the magic number of spacetime dimensions is ten.
If there are ten dimensions in total, we have six extra dimensions! Since we see no evidence of these dimensions in everyday life, they must be tiny (on a scale comparable to the string length), and compact or curled up. Since this theory is supposed to be compatible with general relativity, they should be "flat" in a precise mathematical sense, so their curvature doesn't contribute extra gravitational energy. And to allow for both bosons and fermions, they should be highly symmetric. Such sixdimensional spaces do exist. They're called CalabiYau manifolds: Calabi for the mathematician who predicted their existence, Yau for the mathematician who proved they really are flat.
String dualities
One of the surprising things about string theory, and one of the most interesting from a mathematical perspective, is that fundamentally different assumptions about the setup can produce models of the universe that look identical. These correspondences are called string dualities.
The simplest string duality is called Tduality (T is for torus, the mathematical name for doughnut shapes and their generalizations). Suppose the extra dimensions of the universe were just a circle (a onedimensional torus). A string's energy is proportional to its length; we can't directly measure the length of a string, but we can measure the energy it has. However, a string wrapped once around a big circle and a string wrapped many times around a small circle can have the same length! So the universe where the extra circle is radius 2 and the universe where the radius is ½ look the same to us. The same holds for the universes of radius 3 and 1/3, 10 and 1/10, or generally $R$ and $1/R$.
But what if we want a more physically realistic theory, where there are six extra dimensions of the universe? Well, we assume that the twodimensional string worldsheet is mapping into these six extra dimensions. Our theory will have various physical fields, similar to the electromagnetic field.
To keep track of what a particular field is doing back on the worldsheet, we use coordinates $x$ and $y$. We can combine those coordinates into a single complex number $z$ = $x$ + $iy$. That $i$ there is an imaginary number. When I first learned about imaginary numbers, I was certain they were the best numbers, since they used the imagination; I know that "Why are you wasting my time with numbers that don't even exist?" is a more typical reaction. In this case, though, $i$ is standing in for a very concrete concept, direction: changing $x$ moves right or left, while changing $iy$ moves up or down. If we simultaneously increase $x$ a little bit and $y$ a little bit, we'll move diagonally right and up; we can think of that small diagonal shift as a little change in $z$ = $x$ + $iy$. If you want to be able to move all around the plane, just increasing or decreasing $z$ like this isn't enough. Mathematicians use $\bar{z}$ = $x$  $iy$ to talk about motion that goes diagonally right and down.
Now, back to building our string theory. The fields depend on $x$ and $y$, but they're highly symmetric: to figure out how they act on the whole worldsheet, it's enough to know either how they change either based on a little change in $z$, or based on a little change in $\bar{z}$ (so we don't have to measure rightandup and leftanddown changes separately). If you have two fields like this, they might change in similar ways (both varying a lot due to small changes in $z$, say), or they might change in different ways (one depending on $z$ and the other on $\bar{z}$).
From the physics point of view, this choice is not a big deal. You're just choosing either two plus signs (this choice is called the Bmodel) or a plus and a minus sign (the Amodel). Either way, you can carry on from there and start working out all the physical characteristics of these fields, trying to understand predictions about gravity, and so on and so forth. Because this choice really doesn't matter, it shouldn't make any difference to your eventual predictions. In particular, any type of universe you can describe by choosing two plus signs and working out the details should also be a type of universe you can describe by choosing one plus and one minus, then working out those details.
How do we match up those two types of universes? By choosing different shapes for the six extra dimensions. Using this logic, physicists predicted that if you picked a specific shape for the extra dimensions of the universe and worked out the details of the Amodel, you should be able to find a different shape that would give you the same physical details once you worked out its Bmodel theory.
Now, I said the sign choice wasn't a big deal from the physical perspective. But it's a huge deal from the mathematical perspective. If you only choose plus signs, you can rewrite everything that happens in terms of just powers of $z$, and start doing algebra. Algebra is great! You can program your computer to do algebra, and find lots of information about your sixdimensional space really fast! On the other hand, if you choose one plus and one minus sign, you're stuck doing calculus (a very special kind of multivariable, multidimensional calculus, where experts engage in intense arguments about what sorts of simplifying assumptions are valid).
Thus, when physicists came along and said, "Hey, these two kinds of math give you the same kinds of physical predictions," that told mathematicians they could turn incredibly difficult calculus problems into algebra problems (and thereby relate two branches of mathematics that had previously seemed completely different). Mathematicians call this insight, and the research it inspired, "mirror symmetry."
Further reading
If you would like to learn more about current research in mirror symmetry, here are some resources!

Kevin Hartnett, Mathematicians Explore Mirror Link Between Two Geometric Worlds
Hartnett writes about recent developments in mirror symmetry for Quanta Magazine.

Evelyn Lamb, Why Mirror Symmetry Is Like Fancy Ramen
Lamb describes her interview with Kevin Knudson and me about mirror symmetry.

Timothy Perutz, The Mirror's Magic Sights: An Update on Mirror Symmetry
A more technical description of recent progress in mirror symmetry.