"If there is no one company behind bitcoin, who controls it?" This is the question that many people new to bitcoin ask. What is the best way to convey to them how bitcoin works, and in a way that can be understood by non-techies.
From bitcointalk.org: triple-entry crowd accounting by casascius.
I came up with the term TRIPLE ENTRY CROWD ACCOUNTING as a way to abstractly describe Bitcoin in as few words as possible. I wanted to solicit feedback. This term would make sense to those who are familiar with banking but not with technology.
Slightly expanded (EDIT: and revised):
"Bitcoin is a payment network based on triple-entry crowd accounting. A crowd of computers - run by ordinary Bitcoin users - observes the transactions, produces a single common ledger, and keeps everyone honest. The magic that came from Bitcoin's inventor - the thread that holds the whole thing together - is a documented and published process by which the entire crowd can always agree on what transactions it observed, despite differences in timing and perspective, and even despite varying levels of honesty among participants. Bitcoin's design ensures that no matter how big the crowd, its collective efforts always produce exactly one consistent transaction ledger."
Bitcoin operates based on a set of rules known as the Bitcoin protocol, which is defined by a consensus among its users.
The Bitcoin software is open source so anyone can contribute to its development, but in order for one to have any influence his modified software needs to be accepted by the users (so harmful code will be rejected).
The Bitcoin currency day-to-day bookkeeping is operated by a peer-to-peer network of computers running a Bitcoin software which follows the Bitcoin protocol. Virtually anyone can make his own computer part of this network.