What happens if a player makes a mistake (illegal move) in a tournament? Let’s say I roll 4-3, I move one checker 4 pips and I accidentally move another checker 4 pips instead of 3. What would happen to me? Also, at tournaments - does each backgammon board have a referee?
At a live tournament you will not have a referee watching - and even if there are some other circumstances which have required you to have a TD watching - they are not allowed to tell you of any illegal plays which occur - it is ENTIRELY up to you to watch your opponent and ensure his moves are legal. This is one of the factors which makes live play more demanding - you have to play the game, watch your opponent and ensure you don't mess up either!! In the event of a foul move the rule is this:
A turn is completed when the player picks up his dice. If the play is incomplete or otherwise illegal, the opponent has the option of accepting the play as made or of requiring the player to make a legal play. A play is deemed to have been accepted as made when the opponent rolls his dice or offers a double to start his own turn.
So, essentially, if the illegal play is actually better for you - you can leave it as is (ie if they roll a number which brings them off the bar - but they don't notice it and stay there..), or make them play it - the choice is yours. If you roll your dice, then notice a mistake - that is too late - once you roll, you have agreed their move is ok and must play on.
This is a nice site with the rules quite clearly explained
http://www.bkgm.com/rules.html have a look.