Bitcoin Core is a full node implementation of the Bitcoin protocol with a wallet and GUI. Bitcoin Core is a MIT-licensed Qt 5 GUI application for the original Bitcoin codebase. Additional/previous names include "Bitcoin-Qt", "Satoshi-Client", and "Standard Client". This tag should be used when referring to the Bitcoin Core software.
Bitcoin Core is the rebranding of bitcoin-qt. It is built on top of the original satoshi client and is the current reference client for Bitcoin. It is a full node wallet which forms the backbone of the Bitcoin network as most full nodes on the network run a version of Bitcoin Core.
Bitcoin-qt was officially rebranded as Bitcoin Core in version 0.9.0 to distinguish between Bitcoin-the-network and Bitcoin-the-software. It still retains the prior naming schemes in its user agent and binary filenames.