[chaosite]

So this is Bitcoin 101 and you should really be reading up on it, but basically:

Bitcoins can be divided, the smallest fraction is called a "Satoshi" and is a millionith of a Bitcoin (So a cent is 500 Satoshis). The blockchain is a peer-to-peer, which while technically 3rd party, is not something that "you have coin on". The blockchain is basically a cryptographically signed ledger that says a certain public key has this much bitcoins. If you have the matching private key, you have the coins. Exchanges are mostly used to exchange between Bitcoin and some other currency, like US Dollars or another crypto currency.