The 'green badge' is your licence. There's green, yellow, and 'minicab' licences. Green is a citywide black cab, Yellow is constrained to one area, and minicab is no black cab, but dispatch-only.
'The Knowledge' is the test (and associated training) required to become a black cab driver. It's years worth of study. It's pretty much memorising an A-Z, but also knowing the best routes, at different times of the day, etc.
Three vehicles you can expect to see almost everywhere in London are red busses, black cabs, and scooters with a clipboard on the handlebars. The latter are drivers on their multi-year study of The Knowledge.
https://www.nytimes.com/2014/11/10/t-magazine/london-taxi-te...