What Are Bitcoins?
The rhetoric free definition of Bitcoins is that they're a digital currency, made up entirely of digital coins. While normal fiat currency is backed by the law and word of a government (coming from Latin word "fiat", which means "let it be done"), Bitcoins are backed by the forced scarcity of the code and the word of the creator.
Bitcoins effectively operate in a manner similar to cash. They aren't necessarily tied to a particular person, and exchanges can be effectively anonymous. Individual bitcoins contain code linking them to their user, which is rewritten when the coin is sent out as a payment. The code is only kept by the system for tracking purposes, requiring no actual sharing of information.
Bitcoins are generated through "bitcoin mining." This is done by running either the provided program, or a modified version of it, to provide processing power to the nodes to solve encryption "blocks" and safely complete the pending transactions of others. Tweaks are made to ensure that this process takes roughly ten minutes. The lucky person who manages to find the correct solution will receive a batch of bitcoins. The one significant aspect is that the total number of bitcoins will be capped and held at a strict and arbitrary expansion rate. It will ultimately cap out at around 21 million bitcoins.
This hard limit is appealing to some people who fear manipulation of currency. The value of bitcoins is determined by the market and the market alone.
On a personal note, I don't really see the benefit, as you're effectively trading one devil for another. The market can be far more cruel to fortunes than a central bank, especially when the coins lack a central material backing and are destined for deflation and fueled heavily by speculation.