What is Bitcoin Core?
Bitcoin Core is the reference implementation of Bitcoin. It is an open source implementation of Bitcoin and is not the Bitcoin network itself. Even though 99% of the nodes run Bitcoin Core, a few other implementations can also be used to interact with the Bitcoin protocol. Bitcoin Core is the continuation of the original Bitcoin client released in January 2009 by Satoshi Nakamoto, the creator of Bitcoin.
A client in Bitcoin is the software that facilitates the use of the network. Anybody can download a Bitcoin Compatible client and run it on their computer and interact with the Bitcoin Network. The Bitcoin Core client is a piece of software and as such it has to be maintained and updated; this done through the GitHub repository at github.com/bitcoin, where anybody can submit code for review and if it gets approved by the maintainers of the repository it gets added to the Bitcoin Core client code.
Some people think that since there is a group of maintainers of the Bitcoin Core repository, Bitcoin is centralized; but that is not true, because nobody is forced to use the Bitcoin Core client. Since it is open source anybody can view the code and decide if they want to run it; if for some reason somebody did not like the code and still wanted to run a Bitcoin client they could use any client compatible with the Bitcoin network.
Running a full Bitcoin client is the best way to interact with the Bitcoin protocol, but people often choose to interact with the protocol using light wallets. Light wallets are simpler to use, but sacrifice privacy and do not have a say on potential changes of consensus of the Bitcoin protocol.
The official page for Bitcoin Core is bitcoincore.org, and that is where you can download the binaries for the client. The binaries allow you to verify that the client you run is actually what you were told it was. Downloading and executing binaries is definitely something for more advanced users, but should be done by as many people as possible.