Bitcoin Core is the original BTC client and is available for Windows, Mac and Linux. Core is a ‘full node’ Bitcoin client, meaning that on first-run it will download the current version of the blockchain (currently around 160GB) by connecting to other nodes. It will then continue to download and process data about Bitcoin transactions.
One advantage of this is that it’s much more difficult to link a specific BTC payment address to your identity as Core downloads data about all Bitcoin transactions everywhere. This also protects you against certain types of fraud such as someone trying to spend the same BTC twice, or fooling you into believing you’ve received funds you haven’t actually got.