Bitcoin is a type of cryptocurrency: Balances are kept using public and private "keys," which are long strings of numbers and letters linked through the mathematical encryption algorithm that was used to create them. The public key (comparable to a bank account number) serves as the address which is published to the world and to which others may send bitcoins. The private key (comparable to an ATM PIN) is meant to be a guarded secret, and only used to authorize Bitcoin transmissions.
Network nodes can validate transactions, add them to their copy of the ledger, and then broadcast these ledger additions to other nodes. To achieve independent verification of the chain of ownership each network node stores its own copy of the blockchain. About every 10 minutes, a new group of accepted transactions, called a block, is created, added to the blockchain, and quickly published to all nodes, without requiring central oversight. This allows bitcoin software to determine when a particular bitcoin was spent, which is needed to prevent double-spending. A conventional ledger records the transfers of actual bills or promissory notes that exist apart from it, but the blockchain is the only place that bitcoins can be said to exist in the form of unspent outputs of transactions.:ch. 5
Recall that unique addresses are a proxy for the number of users, because they indicate how many non-zero Bitcoin addresses are in use. This metric has its strengths and weaknesses, which I wrote about here. Also, I have reviewed several other pricing methods which attempt to do the same thing using Metcalfe's Law. You can find those here and here.
In the blockchain, bitcoins are registered to bitcoin addresses. Creating a bitcoin address requires nothing more than picking a random valid private key and computing the corresponding bitcoin address. This computation can be done in a split second. But the reverse, computing the private key of a given bitcoin address, is mathematically unfeasible. Users can tell others or make public a bitcoin address without compromising its corresponding private key. Moreover, the number of valid private keys is so vast that it is extremely unlikely someone will compute a key-pair that is already in use and has funds. The vast number of valid private keys makes it unfeasible that brute force could be used to compromise a private key. To be able to spend their bitcoins, the owner must know the corresponding private key and digitally sign the transaction. The network verifies the signature using the public key.:ch. 5
One of the first supporters, adopters, contributor to bitcoin and receiver of the first bitcoin transaction was programmer Hal Finney. Finney downloaded the bitcoin software the day it was released, and received 10 bitcoins from Nakamoto in the world's first bitcoin transaction on 12 January 2009. Other early supporters were Wei Dai, creator of bitcoin predecessor b-money, and Nick Szabo, creator of bitcoin predecessor bit gold.
The domain name "bitcoin.org" was registered on 18 August 2008. In November 2008, a link to a paper authored by Satoshi Nakamoto titled Bitcoin: A Peer-to-Peer Electronic Cash System was posted to a cryptography mailing list. Nakamoto implemented the bitcoin software as open source code and released it in January 2009. Nakamoto's identity remains unknown.