Oh man. I don't even know where to start. That's partly because there is not one but I dunno, five or so string theories. I'll assume you know a bit about the elementary particles.
Here's what a string theory, in a nutshell is(and I like it for that).
So we built accelerators 'n stuff, and greatly expanded our horizon from the three subatomic particles most people know about. In fact, it went a bit overboard and now there are more subatomic particles then there are elements in a periodic table. That's a lot, mind you!
The only logical(note: there is no hard proof from which to conclude this, it's kind of "this is how it worked out so far" thingy) thing for a physicist to think is that there has to be some simpler underlaying mechanism which gives birth to such complexity. Compare it to language; there are only so many signs in a writing system but you have no problems expressing much more information by simply lining those signs together.
So the scientists assumed that subatomic particles are a tad too big of an alphabet and instead assumed they are in fact words. The natural thing to do was to look for 'the alphabet' of subatomic particles then. And that's basically what strings are.
So here's an overused analogy of what strings are.
Imagine a string. Stretch it and pluck it. It will make a sound. Now stretch it harder. It makes a higher pitched sound now. The sounds in this analogy are what we see as elementary particles. The strings are what actually creates them, depending on how tense they are, in other words, how fast they vibrate. So obviously, one string can play an entire array of elementary particles. That makes the atom an orchestra. Fun, right?
The only thing left to do is push this analogy into as many dimensions as you need to make mathematical sense out of what we observe as the real world and voilá, string theory.
p.s. the last paragraph is what gave birth to several string theories. As you may or may not know, the equations of motion are differential equations and you solve them by choosing the boundary conditions(just some numbers, nothing scary). But what happens is, depending on your choice of boundary conditions, you get different sets of solutions. It's just a mathematical guessing game. Also, you get to choose the dimensionality of space. So far, popular(as in, solved) spaces are 10 and 26 dimensional. In both spaces there are several legit solutions, but this is all math. It's not fit for storytelling.