The bishop strikes diagonally and can only move on squares of its own color. It is a long range piece and prefers open positions and acting on as many squares as possible. The bishop is excellent at pinning pieces to each other and at supporting passed pawns on their way to promotion. The B+Q team is excellent at mating attacks because they can form two types of batteries to shatter a king's shelter (bishop in front and queen in front) and also because they complete each other when delivering checkmate, for example the classical 1.Bd3-h7+ Kg8-h8 2.Bh7-g6+ Kh8-g8 3.Qh6-h7+ Kg8-f8 4.Qh7-f7#.
The B+R team are strong in endgames because both pieces are long range and therefore highly mobile and can quickly shift their attack from one part of the board to another. Together they are also great at supporting a passed pawn, especially a flanking passed pawn, imagine Ra1+a5-pawn+Bg2.
The bishop increases in strength when the center is free of pawns and thereby open for bishop action. The bishop can get trapped behind its own pawns, for example Ba2+b3-pawn+c2-pawn. Sometimes it is enough with one poorly placed pawn to make the bishop bad. The bishop can also be limited in strength by the opponent's pawns. Usually the black light squared bishop experiences tons of problems with becoming bad at some point of the game. Perhaps because white's kingside knight is so important in many opening systems and gradually Bc8 and Ng1 became archenemies.
There are two kinds of bishops. The normal bishop, for example white's Bc4, and the fiancetto bishop, for example g2-g3 followed by Bf1-g2. The fiancetto bishop arises in several opening systems and holds strategic importance. The fiancetto bishop is strong because it has control over the center and it is placed on its longest diagonal (recall that the bishop likes to control as many squares as possible). The fiancetto bishop is also an investment with risk, because after g2-g3 the light squares of the kingside are weakened and in case the bishop is exchanged then these weaknesses can cause problems. The normal bishop can be converted into a fiancetto bishop at any stage of the game. Sometimes black plays Be7-f8 followed by g7-g6 and Bf8-g7 to create a delayed fiancetto bishop.