Public school. It's full name is The Royal College of St Peter in Westminster. Pupils were taught here from at least the 12th century until the Dissolution of the Monasteries. It was re-founded by Queen Elizabeth I. Famous alumni include Ben Jonson, Henry Pucell and Sir Christopher Wren.
Like other ancient public schools this maintains its weird traditions - read this Londonist post for a good introduction.
Credit for this entry to: Alan Patient of www.plaquesoflondon.co.uk