I'm not an amazingly good player, so don't uncritically accept all this...
I would say you can be up on material but losing on position and if you are really losing on position you will lose the game. But, if you are temporarily worse off in position but can equalise the position and remain up on material then you are at an advantage, not only because you can focus more pieces than the opponent in any attack, but also because the opponent cannot easily defuse your attacks by exchanging, as any fair exchange he does will tend to make your material advantage even greater (as a proportion of the remaining pieces).
Of course this only holds up to a point - if you end up materially up, but with insufficient material left to mate the opponent then you have no real advantage at all. To protect against this, if I'm a knight or a bishop up then I try to retain some of my pawns as a backup unless I can see a direct win.
If I get materially up, once I have (if necessary) consolidated so my position is at least equal then I usually like to mount attacks with a view to turning my material advantage into an even greater (ideally game winning) advantage.
Development can be important in determining who has an advantage as well. I've been down a knight or bishop (or even a rook) but my opponent has failed to develop the equivalent piece, so in effect we are on more equal footing. Proper aggressive play (though not overly so) can win a game even with what would appear at first glance to be a decided material disadvantage.
-mike