I'm going to wade into unchartered Slips waters today and head into Bitcoinland. I've been trying to understand Bitcoin from a payment systems perspective, where it has an interesting problem and solution: double spending. The lesson in all of this is how Bitcoin has a sort of built in seniorage--payments are never free. Currently Bitcoin builds in its costs through inflation, which is not particularly transparent, but that will ultimately change to being more transparent--and salient-- transaction fees. By disguising its costs through inflation, rather than through direct fees, Bitcoin effectively incentivizes greater consumer use of the system, much as credit card usage is incentivized through no-surcharge rules preventing merchants from passing on the cost of credit card usage to consumers.
It's hard to describe Bitcoin sufficiently and concisely, but here's an attempt. Cognescenti are invited to correct me; I'm still trying to understand the system. Bitcoin is a completely decentralized peer-to-peer currency and payments network. There is no central Bitcoin authority that verifies transactions, etc. This also means that there is no central for-profit manager that has to be paid to operate the system. Instead, every Bitcoin user has a Bitcoin wallet on a personal electronic device (or webhosted). This wallet consists of open-source software that interfaces with other Bitcoin users' software.