Peter Todd

Bitcoin Core is the main software that runs the Bitcoin network, the direct descendant of Satoshi's original program. It is free and open source, maintained by volunteers and grant-funded developers.

A BIP is a public design document that proposes a change or a standard for Bitcoin. Writing a BIP does not change Bitcoin; node operators decide by running software that implements it.

A full node is a computer running Bitcoin software that checks every block and transaction against the rules by itself. With your own node you don't have to trust anyone about your balance.