Yes, BattleStar Galactica Online is completely free to play – you can create an account at http://tauron.bsgo.com

Its a Space Shooter and Massively Multiplayer Online Role-Playing Game – you will be nagged by pop-ups every once in a while to purchase in-game currency, but in-game currency can also be earned solely through gameplay, so there is no need to ever spend any real money on the game.

I have never actually played the BattleStar Galactica board-game, but I'm vaguely familiar with it, and there isn't much in common beyond the background of the BattleStar Galactica universe.