In a roll to the right, the aileron of the left wing is lowered, thus increasing the camber of the wing and the overall lift it produces. This gives you your roll to the right. The more lift a wing produces, the more induced drag is created.
The result of this extra drag on the upgoing wing results in the nose being "pulled" to the left, or out of the turn.
Overall result of this can be reduced by the use of differential ailerons, but it usually doesn't get rid of the effect all together, so a rudder input is needed from the pilot to maintain a balanced turn.