The Bishop resembles a Bishop's miter (that strange peaked hat which is split down the middle and has ribbons hanging down on either side to tie it on the Bishop's head).
The Bishop is considered one of the minor pieces (it's worth about 3 pawns).
Each player receives two Bishops at the start of the game.
The Bishop moves along the diagonals, forming a giant "X".
Due to the nature of diagonals, the Bishop can never leave the color of the square it starts on. This Bishop is doomed to spend its entire life on just the light squares. It can never land on a dark square. This cuts its scope down to exactly half the squares on the board. As you can see in any one move it can influence a maximum of 13 squares. However, when placed in the corner, it can only influence a maximum of 7 squares.
We need to associate the Bishop with an "X", so think of him as a former Bishop, a defrocked Bishop, an ex-Bishop...
So here is our ex-Bishop, smoking, drinking, and playing cards, though he still wears the miter on his head.
Fortunately, each player gets two Bishops, and each one is placed on a different color square, so that between them they can cover every square on the board.
At the start of the game, the Bishops go on either side of the King and Queen.
It's White's turn to move in this position. What are a few of the possible moves?
(1. Bxa7)
would win a piece, but White has something much better...
(1. Be5#)
Notice how the White Bishops work together: the light-squared Bishop cuts off the Black King's escape square on h7, while the dark-squared Bishop delivers the check. Black's own light-squared Bishop cuts off his King's only other possible escape square.
Back in the previous diagram, if it were Black's turn, what would be his best move?
(1. ... Bd4#) (The same idea in reverse.)