The term board has two meanings in poker:
The Community Cards Edit
Players may speak about cards being "on board" or "on the board", which means that they are present in the community cards. If a player determines at the showdown in Hold 'em that their best five-card hand is actually the five cards in the community cards, that is called playing the board. Players may sometimes need to declare this action, as in "I play the board!", depending on house rules.
More broadly, players are sometimes said to "play the board" when they beat an opponent with board cards of a lower rank than the board cards used by the opponent. This often happens when a pair appears on the board - for example consider a hypothetical situation where a player holds AhJc against an opponent with KdQd and the board runs Ad-Kh-Qh-2c-2s. Both players have two pair, however the player with the ace has played the pair of deuces on the board to complete his superior two pair. This type of playing the board is also called counterfeiting the losing player's two pair.
The Waiting List at a Casino Edit
The list where players sign up to enter a poker game at a brick and mortar casino is called the board. Oftentimes, this is a literal board (usually a whiteboard these days, though it was a chalkboard in the olden days); though more often in small rooms it is a clipboard with some paper on it. Increasingly, the board is computerized, and often displayed on video screens (or using a video projector onto a wall).
The person who manages the board in a casino is called the brush. They are in charge of putting people on the list, and determining who gets the next open seat at a table.