Hmm, I think this is a fine essay for people who have already internalized the notion of algebra per se, but it's not great for a student who has to yet to do so and is asking the titular question "What is algebra?"

Here's my attempt for, say, a relatively smart high school student. Consider this a rough draft; I'm writing it from start to finish without editing. I'd love, love, love feedback, though.

Today even schoolchildren understand arithmetic. We have these things called "numbers." There are different types of numbers like natural numbers, integers, rational numbers, and irrational numbers. We have rules for manipulating these numbers like addition, subtraction, multiplication, and division.

This wasn't always the case, however. I don't mean that humans couldn't always add, but I do mean that it took humans thousands of years and multiple false starts to come up with a sensible way to represent numbers and these operations. Think about trying to do division with Roman numerals, for example. It'd be a nightmare!

There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them and it wasn't obvious what they corresponded to "in real life," if anything. Imagine yourself living in a world like the ancient Greeks, for example, where you represented numbers by talking about lines of a given length. It was very hard for you to talk about numbers per se without drawing a shape that somehow encoded that number. Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?

This is just a story to highlight that although we take our numbering system and arithmetic for granted, this was not the case for most of human history, even most of recorded human history. The way we do arithmetic was an invention that both helped us do arithmetic and helped us understand WTF a number even is.

By the way, if you freak out at the idea of imaginary numbers or the idea of a number i which satisfies the property i^2 = -1, this is no different than the kind of freaking out the ancient Greeks did when they first encountered √2 or other cultures tried to make sense of negative numbers.

Now, let's think about what we really did by inventing arithmetic as we understand it today. You can't point to the number "5" anywhere in the world, right? Even the symbol 5 isn't five per se, any more than "five", "fünf", "|||||", "V", or "五" are five per se. But with one symbol "5" we can represent this abstract thing.

Then something like "5 + 4" might represent the length of a line segment made from concatenating a line segment of length 5 and a line segment of length 4, the age of a 5-year-old in 4 years, the volume of water made from pouring five buckets of water into a pool and then four buckets of water, the number of apples shared between a 5-apple basket and a 4-apple basket, and so on. So, these abstract things we call "numbers" and "arithmetical operations" can represent many more concrete things.

Let's call this process "abstracting." Algebra is what you get when you treat numbers as the concrete thing and apply this same process. With arithmetic, we want to talk about numbers divorced from a particular concrete realization. That is, we want to talk about the number 5 without having to talk about a basket of five apples. With algebra, we want to talk about numbers per se divorced from a particular concrete number.

Remember, when we invented arithmetic we had to invent a bunch of symbols to represent the abstract thing. We do that in algebra, too. Often we use single-letter symbols like x and y, but we could use anything like ☃, ☂, or zorpzop. These symbols "stand in" some number in the same way that the symbol 5 "stands in" for all the things 5 could possibly represent in the world. We can talk about 5 without talking about the things it might represent.

So, we say things like "let x be a number" or "let x be a positive number" or "let x be a rational number." What can we say about x in each of these situations?

For example, if x, y, and z are all standing in for some number, we can say the following:

x + 0 = 0 + x = x regardless of what number x is
x + y = y + x regardless of what numbers x and y are
x + (y + z) = (x + y) + z regardless of what numbers x, y, and z are
x*1 = 1*x = x regardless of what number x is
x*(y + x) = x*y + x*z regardless of what numbers x, y, and z are

These are true because of what we mean when we say "number" and what we mean when we say "addition." It's not as if these are true for some numbers and not all, nor is it as if we know these are true because we've "checked all the numbers." That's impossible because there are an infinitude of numbers.

So, now we might ask things like, "Are there any numbers x such that x^2 + 1 = 0? How about x^2 + x - 1 = 0?" These questions might be hard to answer, but we have now at least invented a language where we can ask them, whereas before "abstracting" arithmetic into algebra we had no easy and succinct way of asking them.

This is no different than not being able to easily ask, "Can we construct an equilateral triangle with side lengths of π?" before abstracting from more concrete things into numbers. Without a symbol for π we have to say things like "the constant that is the ratio formed between the circumference and diameter of a circle." This is how mathematics was done for thousands of years. It was tough going, as you can imagine, and we missed many things that would seem "obvious" to people using our notation.

You can continue this process further, by the way, and abstract further from algebra. This is what mathematicians call abstract algebra (http://en.wikipedia.org/wiki/Abstract_algebra). In this context there are multiple algebras and the "algebra of numbers" becomes the concrete thing in this new system. Linear algebra is a different algebra, for example, with a different sets of "numbers" and a different set of "arithmetical operations" that don't always correspond 1-to-1 with the numbers and operations we find in arithmetic.

Often, when presented with a new physical system of objects that interact in a certain way, we can try to abstract these objects and operations into symbols and derive rules about these abstract symbols and operations that correspond to the workings of the physical system.

We might call this symbolic system "an algebra." For example, Claude Shannon invented an algebra for relay and switching circuits that allows us to understand how they operated and how to combine them without actually building physical circuits. See http://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf