Bitcoin is the name of a new digital currency. You have a file on your computer that says how much money you have, and you can send amounts of your money to other people using the Bitcoin client. Bitcoin has no central authority, but thanks to some advanced math, the files saying how much money you have can't be forged, nor can the transfers be faked or revoked. 
Bitcoins are backed only by the math proving their finitude and the protocol's security. And yet, after a breakthrough of publicity in April and June 2011, the exchange rate of U.S. Dollars to Bitcoins (BTC) has not fallen below $6/BTC . Simplistically multiplying the 7M BTC in circulation by the current price of $10 yields a value of $70M for the Bitcoin network. This value is not purely speculative: Bitcoins are useful and actively being used.
This may be surprising. They aren't shiny (like stone discs) or wearable (like gold or beads), and for ordinary transactions, they don't seem to offer any advantages over existing payment systems. When I'm buying something on Amazon, I'd rather pay in the currency I'm paid in, that my bank uses, with a credit card backed by policies and guarantees and the glory of American capitalism, instead of some weird cyberdollars. As a merchant, the thinking is similar .
Their niche, at least for now, is in transactions that you wouldn't want your current or future government to know about. Because the Bitcoin protocol is anonymous and offers plausible deniability for any transaction, people are using it to buy drugs online, and donate money to radical causes. (As always, illegal ≠ immoral, and you never know what government you're going to have in ten years.) If infrastructure keeps getting built around the currency, it'll be useful in more and more circumstances.
I'm investing in Bitcoins. The main reasoning behind this is purely heuristic: Bitcoin is a useful yet radically novel technology  (like RSA), and radically novel technologies are highly unpredictable. The outcome is bimodal, with a potentially exponential payoff. In other words, the currency will probably fail, but if it doesn't, it'll be worth a lot. And to be explicit, the balance between probability of success and the size of any payoff is impossible to estimate, so there's a very large subjective component here. Shorter-term reasons to invest include that the exchange rate has held steady for a few weeks despite a likely massive selloff by initial holders , and that the supply of new Bitcoins is decreasing.
Since it's already proven useful, the threats to Bitcoin are all from something new happening. These include attacks on protocol, like a vulnerability being discovered, or a massive group of collaborating nodes; legislative attacks against Bitcoin, Tor, or the more general categories they fall into; and the release of a sufficiently superior digital currency that solves the same problems as Bitcoin. My guess is that each of these is more likely than long-term success. But I'm buying anyway, learning as much as I can, and watching closely.
 This explanation is overly simple, but correct enough here. The file is mostly just a private key. See http://en.wikipedia.org/wiki/Bitcoin#Technology to get started. The original paper was published and announced under a mysterious Japanese pseudonym after a year and a half of development in private.
 Though there are multiple exchanges in multiple currencies, there are enough arbitrageurs that using any particular value suffices.
 One advantage for merchants is that transactions are irrevocable. As long as customers aren't interested, though, supporting Bitcoin is not worth their time. The tax implications of Bitcoin are also unclear.
 The essence of the novelty is captured by Hal Finney in the last paragraph of this post from the original Bitcoin announcement in 2008.
 Though likened to a pyramid scheme by some, I have no problem with this redistribution of wealth toward cypherpunks. Early adopters took the risk and did the work to get Bitcoin off the ground.