Bitcoin uses a proof of work system to determine the valid chain of transactions, as well as who gets the initial currency payout. Basically, anyone who controls more than 50% of the CPU power in the Bitcoin network can control the currency.

If you want more details, I'd recommend reading the Bitcoin paper: http://bitcoin.org/bitcoin.pdf

Some of the various forks have tried to work around this by having trusted nodes and the like, but then you lose the decentralized nature and are vulnerable to the trusted nodes going rogue.