Satoshi is the smallest denomination of Bitcoin, and represents one hundred millionth of a Bitcoin. In other words, 1 satoshi = 0.00000001 BTC. Sometimes, "sat" or "sats" is used as an abbreviation for satoshi.
Describing smaller values with satoshis can be much more convenient than describing them with BTC. For example, it's easier to say that a cup of coffee costs 6,436 satoshis, instead of saying that it costs 0.00006436 BTC.