It depends a lot of the position, how many pawns there are, on what colour square they are standing and whether they are stuck on that square or if they can move. Here are a few rules of thumbs:
If there are pawns on two sides of the board, king's wing and queen's wing, it is good for the bishop since it can move fast from one side of the board to the other while it takes many moves with the knight. Thus if there are pawns only on one side the knight has a good time.
If there are many pawns and they are stuck, so the position is closed it is good for the knight, but it makes a big difference on what colour squares the pawns are standing on. If the bishop's colour is the same colour as the squares where the opponent's pawns are standing the bishop can attack them and the bishop is not blocked by its own pawns.
It also matters a lot where the king is located. The king can both defend and attack as well as block the way for the opponent.
In your game where you had a knight and your opponent a bishop we would have absolutely no idea on who would be better off without seing the game, but a draw sounds very plausible.
Hi, sorry if this has been covered. I searched forum posts and din't find anything. My question is this: if the pawns are equal, would the knight rule over the bishop? I would think that they should, since a bishop can only cover one color square. So theoretically if the pawns and king stay off that color, the bishop becomes useless. This should allow the knight to take a pawn or two. Is there a general strategy for these situations? Is my thinking flawed? I recently played a game where I had the knight and my opponent had the bishop. I could only manage a draw, but felt I should have had more. Am I right?