In both of these examples, if you take away the centripetal force (break the string or drive on ice), the object would move in a straight line.

Because of the circular shape of a Ferris wheel, the acceleration in this case is called centripetal acceleration, and its direction is always toward the center of the circle.

You can think of it this way: the force of gravity pulling it toward the Earth is balanced by the centrifugal force (or the centripetal acceleration, which is equivalent *) outward.