A ball travelling at a constant speed in a circular structure is said to be in uniformly circular motion. In such a motion, there is a constant force (called centripetal force) that constantly "pulls" the ball towards the centre of its circular trajectory.
Since the ball experiences a force, it also experiences an acceleration, which has the same direction of the force: towards the centre of the circular trajectory.
But you may ask: why is there acceleration, if the speed is constant?
The fact is that acceleration is the rate of change in velocity of an object, not only in its speed. And velocity includes both speed and direction: therefore, an acceleration can be caused either by a change in speed, or by a change in the direction of motion. In this case of uniformly circular motion, the speed is constant, but the direction of the ball is constantly changing, so the ball is experiencing an acceleration.