What is Bitcoin?
Bitcoin is a virtual currency (in a subset known as a cryptocurrency). Virtual, because it exists primarily online, and a currency, because it's supposed primary purpose is to purchase goods or services. Developed in a 2008 paper by Satoshi Nakamoto (who may or may not actually exist), the currency is unregulated and decentralized.
Bitcoin is unregulated, meaning that there is no central authority creating new bitcoins. However, it is partially regulated by the algorithm by which new Bitcoins are produced. With regular, national currencies, the federal government prints or mints more money in order to regulate inflation and deflation. By doing so, they can regulate the buying power of their currency to optimize international trade and the domestic economy. However, when government printing goes awry, hyperinflation can occur (or some sort of deflation that's bad for the economy, but I don't know the term for that). Government regulation allows for currencies to be more stable, having the same buying power day to day, but also puts the value of your buck in an entity's hand. Without an entity to control the production of new Bitcoins, it is unlikely that Bitcoin experiences runaway hyperinflation, but it is also less stable.
Bitcoin is decentralized, meaning that there is no single server that hosts the data. Instead, all transactions of Bitcoin are sent to everyone running a Bitcoin server and are permanently stored. Server data across all computers are compared and corrected for consistency, ensuring that all clients have the most up-to-date information.
As every transaction is stored, Bitcoins can be stored in wallets, whose value can be easily corroborated by tracing the transactions stored on the servers. These trails are created for each bitcoin (or part thereof), so one can trace the origin of the bitcoin, as well as every single wallet it passed through, on the way to your wallet.
Transactions are done using a public-key/private-key method, which ensures that only the owner of the wallet has access to the Bitcoins stored inside.
The Bitcoin algorithm is based off cryptographical hashing, a way of encoding data that is easy to do one way but difficult to reverse. The most recent transactions are placed into a block and appended to the blockchain (the master chain of blocks), but each block also includes a hash, which, when solved, provides the one who broke the hash with a set amount of Bitcoin. The information that the block has been solved is then propagated through all of the clients so that everybody knows that the block is solved (or mined), so that nobody works on already-mined blocks.
A new block is added to the blockchain approximately once every ten minutes, and the difficulty of the hash is dynamically adjusted to be approximately six blocks solved per hour. The number of Bitcoins in each block is halved approximately once every four years, to prevent runaway inflation of the currency as more people begin mining.
Although made as a currency, Bitcoin seems to have adopted the role of a commodity in the markets, and can be more comparable to gold, whose price is driven by speculators moreso than its actual usage value. Currently used as a currency on only a few major websites, and a few illegal ones, Bitcoin has not seen widespread usage in the marketplace. A single Bitcoin ATM exists in Vancouver, Canada, with a few more planned along the way.
However, as of today, Bitcoin prices have jumped 100%, from $400 to $800, possibly due to the US Congressional Senate hearing where they supported the legitimacy of Bitcoin for financial services.