It means the side who is "on the attack" and can instigate threats which the defender has to answer without generally being able to mount counterthreats of his own.
It does not however necessarily mean having the advantage. One can have a losing game and still have the initiative (at least, for a little while).
The initiative is who is on the attack. In the beginning White has the initiative because he has the first move. But, the initiative can change hands many times during the game. Offense and defense are two sides of the same coin. If you think about it there are two ways to defend against an attack (directly and indirectly). Defending indirectly (your queen is threatened with capture, you make a move that threatens mate on your opponent) usually retains the initiative (you are still on the offense because your opponent must defend against the mate threat before capturing your queen)
I hear teh term initiative used in chess mentor. What exactly are they refering to?