A ball attached on a moving string If there is a ball attached on a string and the string's point of hanging is accelerating horizontally at $\vec{a}$, what will be the forces exerted on the ball that is hanging? It is obvious that there will be a gravitational force downwards and a tension force, and there should be another horizontal force on the ball in the opposite direction of the acceleration of the string, but where does that force come from? It should be from the ball's inertia, but how can that be a force?
 A: The ball rises until the vertical component of tension equals gravity. 
In this stable state the horizontal component of tension is accelerating the ball at the same rate as the vehicle as seen from an external frame of reference. 
In this stable state, to an observer in the car, the acceleration of their frame (car) causes all objects in it to experience a force to the rear. The ball experiences this force, which is balanced by the horizontal tension in the string.
A: 
...and there should be another horizontal force on the ball in the opposite direction of the acceleration of the string, but where does that force come from? It should be from the ball's inertia, but how can that be a force?

This apparent force is known as a fictitious force.  These forces are not present in an inertial frame, but seem to be there in accelerating reference frames.  They are also proportional to the mass of an object.
Imagine that you are doing some experiments with a frictionless table on a train.  You set a mass on the table and at some point in time the train begins to accelerate to the right.
You measure that from your reference frame (the train), that the mass accelerates to the left.  An external (inertial) observer would see that the mass is stationary and that you, the train, and the frictionless table are all accelerating to the right. 
To make $F=ma$ consistent in your frame, you are forced to introduce a bookkeeping or fictitious force on the mass that is directed opposite the train's acceleration.
The same is happening in the car.  The car is accelerating in one direction.  So if you use a reference frame where the car is at rest, it is a non-inertial frame and the fictitious horizontal force on the ball appears.
