As pointed out before, there are many books written about mathematics that are intended for people attending high school, many good ones even.
I always have the feeling that certain types of puzzles are particularly well suited for explaining what mathematics actually is: those where there's a group of prisoners that is allowed to leave prison, once a certain condition is satisfied, but punished heavily when this is not the case. For example:
"In some totalitarian country, there is a prison. On a day, the dictator of the country decides that it would be funny to give a group of 20 prisoners a challenge. The prisoners would be given seperate cells and no means of communication, except for one. There was one particular cell with a lightbulb and switch. Every single hour one of the prisoners would randomly be taken out of his cell and be transported to this particular cell. He might then switch the light on or off. As soon as one of the prisoners would declare that every one of them had been in this special room at least once and if this were correct, all prisoners would be released. However, if it was wrongly proclaimed, something ominous would happen. Before the start of this 'experiment', the prisoners had the chance to discuss their strategy. What should they do?"
The reason to choose this puzzle that is somewhat lengthy to write down, is that I think it contains quite a lot that one could find in "real mathematics". Moreover, lots of people can both understand the puzzle and at least one of its solutions.
What makes me think it's a good way to explain a bit what mathematics is? First of all, people need to understand the puzzle instead of just knowing it. Lots of problems one encounters in maths, one just hears without understanding them fully. After that, there is no designated path that leads to a solution - most people will arrive at the same solution, but in rather different ways. It is not, as it is with most problems in high school, clear cut what needs be done. In solving this puzzle, one needs creative thinking or else one won't be able to solve it. Apart from creative thinking, one of course also needs a strong dose of logic to come to a solution, which of course fails to be the case in most high school problems.
A second aspect that makes it suitable, is that after solving it, one can immediately start asking other similar questions that are not quite the same: what would happen if there was more than one room with a lightbulb? How about more bulbs in one room? Is it still possible if the prisoners wouldn't know the exact amount of prisoners? Etc. Something similar is often the case in maths. One knows a problem, works hard on it and after some time solves it. One then often starts to think of similar problems, possible generalisations of the work.
And then there's another thing: one could, after solving, ask the question if the found method is the most efficient in the sense of time it will on average take for the prisoners to leave the prison. This is somewhat comparable to the concept of elegance in proofs. It often happens that one arrives at some conclusion in a very ugly way, while knowing that there should be a more beautiful or elegant solutions. And then one still keeps thinking about a problem, even when it's solved already.
I'm very interested what others around here think about this - if they agree that this kind of puzzles might help in giving high schoolers an idea of what maths is.