What exactly does game of thrones mean? It is being translated into my native language (Czech) as a game in which one might win one of the many thrones, but to me it sounds more like a game in which different thrones play against each other.
I think part of the confusion is: if there is only one Iron Throne (as there is in the series), why is "thrones" plural in A Game of Thrones. The answer is that "A Game of Throne", "A Game of a Throne", and "A Game of the Throne" all sound really terrible to native English speakers.
Here you should interpret "thrones" as a generic noun, meaning thrones as an abstract concept rather than referring to any particular one. In English, if you have a countable noun, the generic must be expressed as plural (with no article). Possibly, the translation into Czech should have been a game in which one might win the (only) throne, if that phrase can be expressed reasonably succinctly in Czech.
In English the title is ambiguous as to whether the Game is among Thrones, by Thrones, about Thrones, or for Thrones. "Of" is a preposition with very little lexical meaning in English, and in this case it lends a bit of ambiguity. A direct translation to another language might lose that particular nuance.
I suspect that you are seeing it now because of the popularity of George R.R. Martin's "Game of Thrones" book and its release as an HBO miniseries.
The use of the phrase 'game of thrones' is similar to the phrase 'game of chess' but the stakes are much higher- this is a dangerous game played by various people each vying for the throne i.e., competing to be the King and gaining the right to sit on the throne. One thinks carefully before declaring entry into such a game because the consequences of losing is usually death in one form or another.
The Game of Thrones is such a beautiful turn of English phrase partly because of its ambiguity and imprecision. I believe the phrase has several intended meanings, none of which are mutually exclusive.
- Read as the Game of Thrones, it refers to a singular game being played by the Houses of the Seven Kingdoms. In this context it refers to the Houses collective scheming and machinations as they seek to maintain, consolidate, and grow their power, influence, and wealth--at each other's expense. So in this sense it's the game played by the Kingdoms, which are represented by their respective thrones.
- It can be read as a Game for Thrones, which is more specific about the goal of the game: to acquire power (as represented by thrones), and ultimately control of Westeros (as represented by the Iron Throne).
- It is an homage to a similar term from Robert Jordan's The Wheel of Time series, where the various houses and nobels of a particular city play Daes Daemar, or the Game of Houses, which although smaller in scope than the battle for Westeros has several thematic parallels.
- Lastly, game has meaning in relative isolation. Superficially it seems inappropriate, as English typically uses game to refer to contests with lower stakes than death, as in sporting matches or board games. But its use in this context provides insight into the perspective of those who play it. To Cersi, for example, or Lord Frey, the Game is so integral a part of their lives that it does not seem especially violent or brutal. Game in this context is meant to convey the level of desensitization these characters have to situations that we might find shocking or abhorrent.
In the George R.R. Martin novels, I know that the Game of Thrones is mentioned by a character Storm of Swords (and possibly by others in other novels). Here, it refers to the scheming and warmongering that the lords of the Seven Kingdoms engage in. Each throne symbolizes a particular authority.
So what does it mean to play the game of thrones? Well, as the story shows, in order to rule the Seven Kingdoms (to sit on the "meta-throne" that is the Iron Throne) a Lord needs to plan marriages, plot schemes, forge alliances, trick enemies and think of proper strategies. While the rules are not always clear, all the Lords are playing a game, each using the other thrones as pawns.