And most interestingly, you can use your computer to "mine" them.
It's something akin to prospecting for gold, except now you're hunting for 64-digit numbers. By having your computer repeatedly solve puzzles, you're competing with other miners to generate the number that the Bitcoin network is looking for. If your computer generates it, you receive 25 Bitcoins.
At the time of this writing one Bitcoin is worth $42.98, so with correct computations currently worth $1,074.50 a pop there's plenty of motivation to start crunching numbers.
Here's the difficulty, however – the behind-the-scenes math is set up to make Bitcoins increasingly difficult to produce. Once there are 21 million in existence, there can't be any more.