Objects moving in a circle experience uniform acceleration. This force points towards the center of the circle. The ball on a string demonstrates this as it can only travel in one direction with no linear motion toward or away from the center. This force can be experienced in a car if a corner is taken too quickly. This is what is referred to as lateral g-force. With the ball on a string it is the string that accelerates the ball through the tension it experiences.
Centripetal force is the cause of centripetal acceleration. It is the tension in the string that causes the acceleration rather than the force being exerted on it. You can demonstrate this by varying the tension in the string and weight of the ball. You can see that if the string was suddenly cut, the ball would continue on a straight rather than curved path, as the centripetal force would not exist. In astronomy the gravitational attraction causes the centripetal force acting on planets in orbit. If gravitational pull suddenly disappeared, all the planets would float off on a straight rather than curved path.
The most obvious real-world situation to use to explain circular motion is a car taking a corner. If a car tries to take a corner too quickly, it will experience lateral g-force, which tries to drag the car off the road. With the ball on a string, if the velocity increases, so does the tension in the string, but the gravitational weight of the ball does not.
If the weight of the ball is changed and the tension remains static, then another effect is seen. If tension is applied to a light ball, its gravitational weight decreases, giving the impression of increased tension. If the weight increases, the tension in the string decreases; so vertical gravity, the force holding people to the ground, has a greater effect.
Circular motion is an interesting theory, as the combination of weight, tension and velocity do not always act as expected. Try varying the weight and tension of the string and work out the velocity of the ball by timing it around the orbit. This shows why a lightweight sports car can take a corner a lot faster than a heavy truck.