Basics of Material

I expect most anybody who has read a book or two on chess is familiar with the idea of material.  This post really isn't for you, but you are welcome read anyway. 

I think most everybody can recognize that not every piece is equally strong -- partly based on how mobile the piece is and partly in the restrictions of the piece's movement.  Before you exchange one piece for another, it is best to understand who is getting the better deal.

Most modern chess players use a fairly simple technique to quickly judge whether white or black has more material.  They judge the value of each piece based on how many pawns it is worth -- this system isn't exact, but makes it fairly easy to do quick math during a game.  

The quick and dirty values:

I arranged the table from weakest to strongest.  Knights and bishops both get valued at 3 pawns for our pocket calculation -- however they aren't perfectly equal.

Mental adjustments to the final calculation

If you read my post on "Understanding Pawns", you'll see that even pawns aren't perfectly worth one pawn all of the time.  This is really true of all of the pieces.

Pawns are more valuable if they are center pawns (on the D or E file) or if they are on the second to last rank (7th rank for white or 2nd rank for black.)

Any piece grows in value as it becomes more active (see my post "Finding Active Squares") and decreases in value if it is passive.  You generally don't want to trade an active piece for a passive piece.

The Debate on Knights v Bishops

If you have a firm opinion on this debate, nothing I say will change this, however I think it's fairly common for most people to give bishops a slight edge in value over knights.  So, you might see that one player has two knights and the other has one knight and one bishop and give a hair worth of advantage to the player with the bishop.  The actual comparison though really depends on different factors that relate to how effective the different pieces will be.

Knights grow in value in closed positions or positions where all of the pawns are on one side of the board.  In a closed positions where mobility is restricted, the knight's ability to jump over other pieces is valuable.  In end games where all of the pawns are on one side of the board, a lone knight's ability to reach every square color gives it an edge over a lone bishop which is restricted to a single color.

I should also note that in terms of exchange value, a white knight on the fifth rank is more valuable than a white knight on the second or third rank.  The same goes for a black knight on the third rank being more valuable than a black knight on the sixth or seventh rank.  Because knights don't move across the board very quickly, a knight that has found an advanced outpost can be very valuable. 

Bishops are more valuable in open positions where they can move freely about.  

A bishop pair is very strong in an endgame.  When one player has a pair of bishops and their opponent does not, the player with the pair can add almost a whole pawn worth of material to their count.  

A lone bishop in an endgame is best in situations where pawns are on both sides of the board and the ability to move quickly is more important than the ability to reach every square.  There are situations where a bishop can control a diagonal in the endgame and stop many pawns at once from advancing.