When used properly, bishops can be quite powerful. In many positions, a bishop can prove to be much stronger than the other minor piece, the knight.
Bishops Like Open Diagonals
Open positions, where pawns -- especially central pawns -- have been traded, tend to increase a bishop's potential. Place bishops on open diagonals, where they can exert control over as many spaces as possible.
The diagram above comes about in a variation of the Danish Gambit -- the moves played were 1. e4 e5 2. d4 exd4 3. c3 dxc3 4. Bc4 cxb2 5. Bxb2. The letter/number combinations here represent positions of pieces on the chess board as well as the specific moves a player makes with those pieces. For example, the capital "B" stands for the "bishop" piece, the lowercase letter-and-number combinations, such as "e4," represent the positions of the pieces on the board, and the "x" shows that a piece has captured an opposing piece by moving into a particular spot on the board. In this case, white has sacrificed two pawns but has compensation due to the two very strong bishops he has developed while Black was busy taking pawns. You can check out the algebraic notation for chess pieces and moves on this web page published by FIDE, the international chess federation.
While opening theory says that the position above favors Black -- two pawns is a little too much material to give up, even given White's big lead in development -- White's bishops are dangerous attackers thanks to the long, open diagonals they've been placed on. Black must defend accurately to retain his advantage.
Good and Bad Bishops
Bishops can be classified as "good" or "bad" based on their relationship with their pawns.
If most of your pawns -- particularly the central pawns -- are on the same color squares as one of your bishops, that bishop is considered a "bad" bishop. Similarly, a bishop that does not share the same color as most of your pawns is considered a "good" bishop.
In the diagram above, both players control a light-squared bishop. As White's pawns are on dark squares, his bishop is good. Black's pawns reside on the same light-colored squares that his bishop moves on, making his bishop bad.
While these names are commonly used, they do not necessarily reflect how effective a bishop might be in a given position -- they are simply a way of describing the piece. That said, good bishops are often more advantageous than bad ones. Good bishops have more freedom of movement, and control squares that their allied pawns cannot. Conversely, "bad" bishops can sometimes be useful, as they and their pawns can defend each other.
A bishop that is outside of its pawn chain is an active bishop. Active bishops have greater freedom and are generally better placed than those still trapped inside the pawn chain. Either "good" or "bad" bishops can be active.
In the diagram above, both White and Black have made their bishops active by developing them outside of their respective pawn chains. Notice that while Black's bishop is technically "bad," it has taken a strong post at d4 and has plenty of scope for movement.
Bishops of Opposite Colors
Because bishops are forced to stay on squares of a single color, they have some interesting properties that set them apart from other pieces. For instance, both sides may be left with just one bishop -- with one side retaining its light-squared bishop, while the opponent has his dark-squared bishop.
In the middlegame, these opposite-colored bishops can become strong attacking weapons. As neither bishop can directly confront the other, it is difficult to use them in defense when the other player's bishop is attacking. In this sense, having bishops of opposite colors gives the attacking player a material advantage.
In the endgame, opposite-colored bishops tend to benefit the weaker side. Typically, it is possible -- and often quite simple -- to secure a draw when losing by a pawn or even two in an opposite-colored bishop endgame. The defending side can set up a blockade on the squares patrolled by its bishop, and the stronger side cannot use its bishop to break this defense.
In the diagram above, Black is ahead by a pawn and appears to be very close to promoting his pawn. However, the presence of opposite-colored bishops makes this an easy draw for White. Black cannot remove the White bishop from the a1-h8 diagonal, nor can Black's bishop block the diagonal to help his pawn promote. If Black ever attempts to promote the pawn, White can capture the pawn with his bishop; even if the bishop is lost, the game will be a draw, as Black cannot force checkmate with just a king and bishop.
Bishops in the Endgame
Bishops are strongest in endgames with pawns remaining on both sides of the board. This situation allows them to use their long-range capability to its fullest and minimizes the handicap of only being able to access one color of squares. This is contrasted with the other minor piece, the knight, which excels in endgames where all the pawns remain on one wing because it can cover squares of both colors.
In the diagram above, the White bishop is using its long-range abilities to its full potential. While Black has five connected passed pawns, the White bishop stops all of them by controlling the long diagonal. White will win easily by promoting his only remaining pawn.
Bishops in the Endgame: The Wrong-Colored Bishop
Sometimes, even having an extra bishop and pawn is not enough to win in an endgame. This occurs when the pawn is a rook pawn -- meaning it is on either the a or h file -- and the bishop is not on the same color as the square on which that pawn would promote.
The diagram above illustrates this type of endgame. White's pawn on a7 would like to promote to a queen on a8, a light square. Unfortunately, White only controls a dark-squared bishop, making it impossible for the bishop to help protect a8 or drive the Black king away from there. Even though it is White's move, there is no way to make progress; either White may move his king away and allow Black to shuffle his king between a8 and b7, or White can play a bishop move and stalemate Black's king.