My style of play usually values the pieces:
N=3 pawns
B=1.75 pawns
R=2.5 pawns
Q=6 pawns
P= 1 pawn
If my opponent is unable to value the pieces correctly and makes the wrong trade, I usually see it as a mistake.
So I assess my opponent's skill by seeing if they can value the pieces right (and therefore make the right trades) as the position changes.
Much has been made of best responses during openings and such. However, what do you look for as weaknesses in your opponent's moves/positions? How do you assess your opponent's ability as you progress through the game, aside from rating?