The Collaborative International Dictionary of English v.0.48:

backgammon \back"gam`mon\, n. [Origin unknown; perhaps fr. Dan.
   bakke tray + E. game; or very likely the first part is from
   E. back, adv., and the game is so called because the men are
   often set back.]
   A game of chance and skill, played by two persons on a
   "board" marked off into twenty-four spaces called "points".
   Each player has fifteen pieces, or "men", the movements of
   which from point to point are determined by throwing dice.
   Formerly called tables.
   [1913 Webster]

   backgammon board, a board for playing backgammon, often
      made in the form of two rectangular trays hinged together,
      each tray containing two "tables".
      [1913 Webster]

WordNet (r) 3.0 (2006):

backgammon board
    n 1: the board on which backgammon is played