A satoshi (or "sat") is the smallest divisible unit of Bitcoin: 1 bitcoin equals 100,000,000 satoshis, similar to how a euro is divided into 100 cents, though with far more decimal places of precision.
Why this unit exists
By being divisible into such small units, Bitcoin allows for very small payments without depending on the price of "one whole bitcoin" being a manageable figure — something especially relevant now that the price of a bitcoin is a large amount in euros or dollars.
The origin of the name
"Satoshi Nakamoto" is the pseudonym of the person (or group of people) who published Bitcoin's original technical paper in 2008 and built its first implementation; their real identity has never been publicly confirmed. In their honor, the network's smallest unit bears their name.
Common usage
Many exchanges and wallets let you display balances in satoshis instead of bitcoins, which is especially useful for small holdings, where expressing the balance in bitcoins would give a figure with many leading zeros.