how do I know if the piece is important to me or not

Assuming you know the piece values (queen 9, rook 5, knight and bishop 3) (these values are just general guides and imprecise at best, but good for beginners) and mean that you want to know when it is a good idea to trade a piece for a piece, think of it this way - would you trade an archer on a hill for an archer in a ditch?  Would you trade an archer on a hill for a swordsman at the bottom of the hill?  In other words, you want to look at what each piece can actually do -- the squares the pieces can control, how active it is or can be, whether it can attack key squares or guard weaknesses.  Sometimes there are other more advanced issues not directly related to what the particular pieces are doing -- for example, if the exchange of pieces will introduce a weakness when your opponent recaptures.  But in general, look at the activity of the pieces you are thinking of trading.  Hope that helps.  

While playing chess we often reach the position, where without thinking further we just exchange the piece. But before trading/exchanging ask your self the following questions

1.) Whether after the finishing of forcing line that involves trade, my position/activity will be better or not. In this case you will have following options/answers

1(a) Only your activity is increased. Then you should go for trade/exchange. Exchanging your bad piece also comes under this category.

1(b) Only your opponent's activity is increased. You should never exchange at that moment.

1(c) Activity of both players increased. Then you have to evaluate that what else more beneficial you are getting. In case you find out some thing more beneficial then go for trade otherwise don't trade and find some another move.  

@PET_PET_LOU

As White, would you trade your bishop for the knight in the position? Explain your decision.

I answered it. Didn't realize it was for OP. 

@Pawnlings

I didn't manage to read your comments in time, but I presume you are correct, and I would presume that almost every good player knows what to do.

@PET_PET_LOU

One more for you. How about this, with White to move? Would you swap bishops?

Does trading the piece improve/hurt your position or your opponent's?  For example, if the trade allows you to later occupy an unassailable Knight Outpost or if you're creating Doubled Pawns for your opponent it may be worth the trade.

That's the basic question you must answer when you trade.