Nothing is stored, actually.
Here, on Earth we are used to the fact that things "naturally" stop moving after a while (if we do not make them continue the movement somehow) "by themselves". But this is only apparent, as there is always certain force that prevents movement. This force is gravity often coupled with friction - gravity pulls the ball and the surface prevents further movement down (toward the center of Earth), but also - since the ball is "squeezed" by the gravity and the surface - it prevents the movement forward (parallel to the surface). It's like squeezing the ball inside a tube that's smaller in diameter (or rather between two surfaces).
That's the situation we are used to. We consider it normal, because we observe such behaviour everyday, since the childhood.
However, if there is no force opposing the movement (which we never experience on Earth) the ball will just keep moving. Because ... why shouldn't it? If there is nothing on the way, and no force working on the ball, it will just maintain the state it is in, i.e. movement.
This is what is known under the name of Newton's First Law of motion.
Why doesn't a force stop moving the ball immediately? If you place a wall on the way of a moving ball it will stop immediately. What about a force, like gravity?
Well, movement is always relative (to something). When you say a ball is moving, you usually assume it is moving with respect to Earth. So when you are sitting on a bench in a park and put a ball beside you, you will say it is not moving. But when you are on a train and have a ball with you, and put it beside you ... is it moving or not? Right, the answer depends on what you assume as the point of reference - the train or, say, a tree you are just passing by.
So, back to the question, everything depends on the relative movement of the observer. If he moves along with the ball and suddenly a force starts exerting its influence, he will say the ball started moving toward the force immediately. But if he sees the ball as moving (and the initial movement of the ball will be away from the source of the force), then he will see things differently - he will say it slowed down first, and then, after a momentary halt, begun to move toward the source with acceleration.