The alleys of a tennis court are the long rectangles to the very left and right of your court.
This picture will give you an idea of what I mean by the very long rectangles on the left and right:
http://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Tennis_court_imperial.svg/292px-Tennis_court_imperial.svg.png
They run along the long side of the court.
They are only used when playing doubles, that is 2 players versus 2 players. This is to allow more court room for more players to hit the ball.
During singles, 1 vs. 1, the alleys are ignored and any balls that land in the alley are considered out.
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The portions of the court on each side that is in play in doubles and out of play in singles are known as the doubles alleys.