This is the question that every new student I have asks after a few games played. I've wondered this myself. Through my two decades of playing the game I've read a great deal on the history of the evolution of the board itself.
Today, in general, we play on boards that are 19x19, but this wasn't always so. In Ancient times Go was played on boards from 15x15, 17x17, and 19x19. Somewhere around the 200CE we see everyone settle out on the 19x19 board.