Why does a ball bounce? If an object is acted on by equal and opposite forces then it will be in equilibrium, and it's acceleration or velocity (and so direction as well) will not be changed.
So when a ball bounces, it exerts a force on the floor, which matches the magnitude of the force in the opposite direction (the ball is bouncing perfectly vertical), up. So how is it's velocity/direction changed? If the forces are equal and opposite to each other. In order for it bounce, surely the force acting from the floor to the ball must be greater than the force acting from the ball to the floor?
 A: You've misunderstood the statement.
When A exerts $\vec{F}_{A \to B}$ on B, B exerts an equal and opposite force $\vec{F}_{B \to A} = - \vec{F}_{A \to B}$ on A.
The only forces acting on the ball a gravity and the normal force, and the floor experiences a force from the ball which is equal in magnitude and opposite in direction from the normal force on the ball.
A: The easiest way to understand this is by flows of momentum, the ball has momentum down, and when it hits the floor, it transfers twice this amount of down momentum to the floor, and so is left with negative this much down-momentum, which is up-momentum.
The statement of Newton's laws are nothing more than the flows of momentum from object to object, and how the object moves when momentum is building up on it. Understanding it this way is helpful for clearing up the elementary confusions. This is a supplement of dmckee's answer, which is correct, but it doesn't give the intuition of momentum-flow, which makes the misunderstanding impossible to formulate.
A: The problem is a common conflation of 1st and 3rd law forces. Yes, the normal force (from the floor up) and the contact force (from the ball down) are equal and opposite since they are 3rd law forces. However, to understand why the ball's motion changes we must focus on external first law forces. If the the normal force up exceeds the downward forces of Earth's gravity and drag from the air, the ball will accelerate upward.
