As beginners, we learned the “static” values of the pieces: Knight equals three pawns, Bishop equals Knight, Rook equals five pawns and so on. Based on these values, we knew whether we were about to make a good trade or a bad trade just by counting. Later, we learned more sophisticated evaluations: bishops are usually better than knights in open positions (definitely with pawns on both sides of the board); Knights are better than Bishops in congested positions, much like motorcycles maneuvering deftly in congested traffic. We also learned a bit about the art of sacrifice – giving a rook for knight and pawn can be quite OK if you get something additional in the bargain, e.g. the opponent’s weakened pawn structure in the vicinity of the King.
It is quite common to see a piece sacrificed for a couple of pawns. Of course these need to be pawns of some importance, e.g. pawns shielding the King from attack, or pawns preventing your pawn from promotion, or center pawns whose absence allows your pawns to come rolling through the center, unopposed. In such cases, the defending side would be relieved to give a piece back to stop the invading hordes.
It is more rare to obtain many pawns for a piece. Such cases are not really sacrifices, rather, gains in material. I was surprised, therefore, in a recent tournament to twice have games in which I obtained four or more pawns for a piece! In the first game below I had promising chances, but time shortage against a strong opponent prompted me to chicken out and take the perpetual. While I feel some shame for this, I take pride that in time pressure, for once, I made a “rational” decision. In the second game, with five pawns for a knight, it looked like i would have an easy time of it. But my opponent fought back resolutely, partially blockading my (many) pawns and creating tactical threats with the extra piece. The turning point came at move 37 where we both missed the shot 37…Nf1+!!, which would have forced a draw.