Bitcoin Mixing service provide the ability to exchange your bitcoins for different ones which is not related with the real owner. It is a service where used to mix one's funds with other people's money, designed to puzzled the track back to the funds' original source.
Vanity address is a normal bitcoin address that starts with some letters/string of characters that suits to you. In other way it's a customized bitcoin address that something from your name or screen name, code name etc.
Bitcoin is a form of digital currency, created and work electronically. Nobody owns and controls it, it is an open source designed as public. It’s the first decentralized peer-to-peer payment network that is powered by its owner or its users that no central authority.