The speed of the air moving past the ball's surface is the trick to a curveball. As the ball spins, its top surface moves in the same direction in which the air moves. At the bottom of the ball, the ball's surface and the air move in opposite directions. So the velocity of the air relative to that of the ball's surface is larger on the bottom of the ball.

   The higher velocity difference puts more stress on the air flowing around the bottom of the ball. That stress makes air flowing around the ball break away from the ball's surface sooner. The air then travels at the top of the spinning ball, subject to less stress due to the lower velocity difference, can hang onto the ball's surface longer before breaking away.

When a pitcher releases a curveball, the pitchers grip will put a spin on the ball. As the ball moves through the air it experiences a force called drag or air resistance.

If the ball isn’t rotating the drag will only make it go slower. For a baseball, when it is rotating, the force of the drag is different at different points of its path. The air friction is less on one side of the ball and because of that the ball tends to curve.

Something else that helps a baseball to rotate or spin is the roughness of the ball. If the stitches on the baseball weren’t on the ball the ball would be really smooth. If that were the case any of the special pitches such as the curveball wouldn’t be possible.  For every action there is an equal and opposite reaction. So, as the spinning ball throws the air down, the air pushes the ball up in response. A ball thrown with backspin will get a little bit of lift.

   "A major league curveball can veer as much as 171/2 inches from a straight line by the time it crosses the plate. Over the course of a pitch, the deflection from a straight line increases with distance from the pitcher. So curveballs do most of their curving in the last quarter of their trip. Considering that it takes less time for the ball to travel those last 15 feet (about 1/6 of a second) than it takes for the batter to swing the bat (about 1/5 of a second), hitters must begin their swings before the ball has started to show much curve. No wonder curveballs are so hard to hit." (http://www.exploratorium.edu/baseball/)