The phrase "scientia potentia est" is a Latin aphorism meaning "knowledge is power". It is commonly attributed to Sir Francis Bacon.
The Queen addresses three corgis by name: Honey, Sugar and Pippin. All three existed in real life, and Sugar was photographed with the Royal Family for the cover of Australian Women's Weekly in 1959.
Elizabeth and her tutor make several references to Walter Bagehot (pronounced Bajut), editor of the "Economist" from 1861 to 1877. He was Britain's leading constitutional scholar and introduced the distinction quoted in this episode: that the Monarch is head of the 'dignified' parts of the constitution (those that "excite and preserve the reverence of the population") whilst the prime minister was head of the "efficient" parts ("those by which it, in fact, works and rules"). This is a "distinction [that] is often drawn, even today" per the Economist website. And, emphasising its relevance today, the Economist's U.K. politics columnist writes under the name "Bagehot".
The episode's title ("knowledge is power") has a double meaning in the context of the episode, referring both to Elizabeth's desire for an education in world affairs, and to her knowledge of specific information about which she is being kept in the dark by her government.
The scene showing Eton College was actually filmed at Winchester College.