This is the only episode in the series where the phrase "game of thrones" is spoken.
Charles Dance (Tywin) learned to skin a real stag in his very first scene. Tywin never does this in the books: in the first novel 'A Game of Thrones', the character Randyll Tarly is introduced this way during his son Sam's flashback. Since Randyll doesn't appear in the show until season 6, the scene was used for Tywin's introduction. According to D.B. Weiss and David Benioff, the scene was shot for about 50 takes. Apparently, Nikolaj Coster-Waldau enjoys venison and hoped that the stag could be sent to the catering department, but that never happened.
The episode title is part of a quote Cersei says to Eddard: "When you play the game of thrones, you win or you die. There is no middle ground." This catchphrase has been frequently used during the promotion of both the novels and the TV series.
Tywin's act of butchering a stag (the sigil of House Baratheon) is perhaps a foreshadowing of Robert's death, caused by Lannisters.
In the throne room when Joffrey is first proclaimed as the new king following Robert's death, he is referred to as "Joffrey of Houses Baratheon and Lannister", and is the only character in the series to be referred to as belonging to two houses simultaneously.