The little pieces of each bitcoin are referred to as "satoshis". There are 100 million satoshis per bitcoin, so up to about 2.1 quadrillion satoshis can exist.

When people talk about "buying a bitcoin", they're just using that as shorthand for "buying 100 million satoshis". People don't usually actually trade whole numbers of bitcoins. For example, today I sent 1.12952196 BTC (or 112,952,196 satoshis) from my MtGox account to my Electrum wallet.

Other shorthand labels that people use include:

• Bitcent - 0.01 BTC, or 1 million satoshis
• Millibitcoin (mBTC) - 0.001 BTC, or 100 thousand satoshis
• Microbitcoin (uBTC) - 0.000001 BTC, or 100 satoshis

In general, people will use whichever labels they find convenient.