I try here to describe aspects of the protocol with emphasis on how to pay and the security architecture.
It is radically open. Not only is there open source software to run it, the transactions are visible to all but the identities are nominally opaque. With open transactions cluster analysis quickly identifies players, even if not by their real name. Their bitcoin identity may be more relevant, however, than what is on their birth certificate. See this.
Links: Wikipedia, The Economist (remarkably technical!), Brian Warner’s slides, Variants
Eric Hughes was studying audible banks 10 years ago with crypto protocols involved. I don’t recall what problems he was trying to solve; I think it was to avoid having to trust banks.
how it works
Whenever you spend a Bitcoin, you cryptographically sign a statement saying that you have transferred the coin to a new owner and you identify the new owner by their public crypto key.
Original Paper; research; current price; Developer Notes by Krzysztof Okupski.