When the ball swings in the air it is, basically, referred to as swing bowling. The theory about swing-bowling is that when the ball, with one of its surface smooth and the other rough, moves through the air, the aerodynamics applied on the ball makes the smooth surface cut faster through the air and hence makes the ball swing either out or in - depending on which side the smooth surface is facing
Now, reverse swing is when the ball starts behaving otherwise. That is the ball starts swinging in the opposite direction. For example: for outswing the shine of the ball faces the on side and swings away from the batsman (see outswinger for detailed explanation of outswing) but in the case of reverse swing, with the shine facing the on side the ball swings in to the batsman.
The theory behind reverse swing is that when the ball gets older and softer and when the shine on one side of the ball is maintained and the other side gets very rough and scuffed up (due to ware and tear over a long period of time) the ball starts reverse swinging when delivered at high speed (above a specified speed).
Reverse swing is a phenomenon that has evolved towards the later years of cricketing history. Pakistan cricketers are reckoned as the pioneers in the art of reverseswing. Wasim Akram, Waqar Younis, Glenn McGrath are few of the cricketers in contemporary cricket who have mastered the art of reverse swing.