In general one would hope that the word "chess" would apply solely to WesternChess, with other variants mentioned specifically by name...
The ChessBishop is an interesting piece in that it is restricted to a single color from the start of the game. Because of this, two opposite-colored ChessBishops? are generally considered more powerful than two ChessKnights in open positions, in that they are long range and can attack all squares on the board (each can cover half, determined by its square color). In closed positions, the ChessBishop is often reduced to a glorified ChessPawn.
As a long range piece, the ChessBishop's strongest value generally applies to pins and skewers, although it is also capable of forking in certain circumstances. The best way to make a ChessBishop "happy" is to place it on a long diagonal, unopposed by your opponent's like-colored ChessBishop or pawn chains of either color. Generally, a ChessBishop blocked by a pawn chain is much less useful than one that has a freer range of motion, and is said to be "biting granite." Continuing on this note, a ChessBishop can often "protect" an entire pawn chain by "attaching" itself to the ChessPawn at the base of the chain. A ChessBishop and ChessPawn can take part in a form of "mutual protection" which can make them very difficult to capture later in the game.
One must carefully consider entering an endgame with only ChessPawns and ChessBishops because a small material advantage (such as one extra ChessPawn) may not be enough to force victory. In fact, endgames where one player has one ChessPawn on a rook file and a ChessBishop uncapable of guarding the queening square and the other player has only his ChessKing are almost always drawn. Even with an advantage of a ChessPiece and a ChessPawn, victory is not ensured. (This is only the case when the ChessPiece is a ChessBishop. This weakness must always be kept in mind before entering the endgame.)
Almost-even endgames with ChessBishops on the board are even more frequently drawn when each side has one bishop and the two bishops move on opposite-colored squares. These are typically known as endgames with "bishops of opposite color" and the "drawing power" of such endgames is such that in some cases even centralized connected passed pawns are not enough to force victory for the stronger side.
When paired with a ChessRook?, a ChessBishop can help create a "windmill attack". This series of moves is very uncommon and so powerful that its creation often leads to an immediate surrender. (Note: I've been playing chess seriously for about four years and have never played in a game where a windmill attack occured. I've seen it threatened by myself or an opponent, but it has not once actually come up. That's how rare it is.)
While on the ChessSide, all other things being equal, a ChessBishop on the board is roughly equal in power to a ChessKnight on the board, on the CheckersSide dark-squared bishops tend to die very quickly. Since pieces may not be dropped on light squares, a bishop in hand is usually destined to be transferred back to the other side of the board.