Board control is an unofficial term usually referring to a player's ability to keep the opponent from building up minions on the battlefield, or "board". Control decks are named after this strategy, aiming to control the opponent's minions, prevent them from gaining too much of an early advantage, and winning in the late game.
Board control is often accomplished by playing one's own minions and trading them effectively, so the term is sometimes used to mean that the player is also able to keep their own minions in play, in addition to removing the opponent's. However, the correct application of removal cards may accomplish the same overall goal, even without playing many friendly minions.
Board control is strongly associated with tempo - a player that has played many cards efficiently compared to the opponent generally has more control over the state of the board than that opponent. However, it is also possible to control the board in a reactive, defensive manner, in which case the goal is simply to ensure the opponent does not build up significant tempo, rather than actually gaining it oneself.
For a deeper discussion of tempo and board control strategies, see Tempo and Control deck.