How can an object with zero potential and kinetic energy ever move? I am not sure how to ask this question but I am learning about potential energy in my high school physics class. From the definition of potential energy, (energy stored in an object with the potential to convert into other type of energy), I don't understand how an object (let's say a ball) on the ground, which has zero kinetic energy and zero potential energy, can fall off a cliff  and gain kinetic energy when it had no potential energy. How can it now be gaining energy? Also, another question that was already discussed in SE (but I didn't find my answer there), is why do we talk about the potential energy of a system (ball+Earth) but kinetic energy of an object (ball)? 
Thanks.
 A: Where we define the potential energy to be $0$ in classical mechanics is arbitrary. All that matters is the change in potential energy. Since you are just learning this stuff I will assume you are in an algebra based physics class, so I will avoid using calculus here.
Potential energies are nice because they tell us how much work is done by a conservative force. More specifically, the work done by a conservative force is given by
$$W_{cons}=-\Delta U$$
where $U$ is the potential energy associated with that conservative force. This is useful because we also know that the net work done on an object determines its change in kinetic energy
$$W_{net}=\Delta K$$
So, if we consider your case where we just have one conservative force acting on the object, we can conclude that
$$\Delta K=-\Delta U$$
And so we see here that the only thing that determines how the motion of our object changes is just the change in potential energy. If we define the zero-point to be at the top of the cliff, then as the object falls its kinetic energy will grow and its potential energy will decrease and be negative when it hits the bottom of the cliff. If we define the zero-point to be at the bottom of the cliff then as the object falls its kinetic energy will grow and its potential energy will decrease to $0$ when it hits the bottom of the cliff. In either case the same thing happens because we have the same change in potential energy.

Also another question that was already discussed in SE (but i didn't find my answer there) is why do we talk about the potential energy of a system (ball+Earth) but kinetic energy of an object (ball)?

Typically in introductory physics classes we just consider the ball being in a uniform gravitational field and we don't even consider the Earth. However if you want to include the Earth in your system then when the ball falls the Earth will actually move slightly upwards to meet the ball. It will still be the case though that the work done by gravity on each object is related to the change in the potential energy of each object, which results in a change of kinetic energy.
A: The energy of an object consists of its internal energy (kinetic and potential energy possessed internally to an object at the atomic or molecular level) and external kinetic and potential energy. Its external kinetic and potential energy is always with respect to some external frame of reference. 
The ball on the surface of the earth may have zero potential energy with respect to the surface of the earth, but has gravitational potential energy with respect to the center of the earth. It doesn’t move because the force of gravity downward on the ball is equal and opposite to the force exerted upward on it by the ground. But imagine if there was a trench that went from one side of the earth to the other. If the ball were positioned over the hole and allowed to fall, it would accelerate toward the center of the earth losing potential energy and gaining kinetic energy, the latter being a maximum at the center where its potential energy is zero. Then it would decelerate as it goes past the center as it gains potential energy and loses kinetic energy, ultimately stopping at the other side (ignoring air resistance).
Even the kinetic energy of the object is relative to the frame of reference where the velocity is measured. If you are in a car moving at velocity $v$ with respect to the road, it has zero kinetic energy in your frame of reference. It is not “moving” in your frame of reference, but is moving in the frame of reference of a person standing on the road. To that person the car has kinetic energy. The person standing on the road is “moving” with respect to your frame of reference in the car, so that person has kinetic energy with respect to you in the car. The person standing on the road has kinetic energy with respect to a frame of reference other than the earth (since the earth is rotating). And so on…
Bottom line, everything can be considered to have kinetic energy (be moving) with respect to some frame of reference and have potential energy  (due to its position) with respect to some frame of reference. 
Hope this help.
A: 
I am not sure how to ask this question but I am learning about
  potential energy (high school physics) and from the deffinition of a
  potential energy (energy stored in an object with the potential to
  convert into other type of energy)

this definition of potential energy, sounds more like the energy itself. More below

i don't understand how for eg. when we have an object (let's say a
  ball) on the ground, it has zero kinetic energy and also zero
  potential energy and now let's say the ball starts falling of a cliff
  so it will be gaining kinetic energy but when it had no potential
  energy, how can it be now gaining energy?

indeed,  in the planet+object system, the true 0 potential, would be better put at the center of gravity the earth if the reference frame is attached to the earth, if the reference frame is attached to the object, then it should be on the center of gravity of the object, and if the reference frame is neither, the 0 potential should be at the center of gravity of the object+earth system. 

Also another question that was already discussed in SE (but i didn't
  find my answer there) is why do we talk about the potential energy of
  a system (ball+Earth) but kinetic energy of an object (ball)?

Because you need to define what type of potential energy we are talking about, in this case it's gravity (could be electrical for example)
We now see that potential energy is relative to a type of force (gravity), reference frame, and the final nail in the coffin of your definition of potential energy, is that it's not "stored in the object" . Potential energies depend on the spacial position of the object in the force field.
I hope all these elements will help you understand potential energy. Meanwhile your teacher probably gave you a simplified version of this that is enough to solve the problems you are supposed to solve for now. The main idea the teacher probably wants to convey is that energy is conserved, and that like voltage, we are more interested in differences in potential than absolute potentials.
A: A moving object, due to its inertia,  has an energy (that could do damage in case it impacts something :) We call that energy "Kinetic"
A system (like the earth and ball) can potentially (if allowed) cause one or both objects to move because of the gravity (in this case).  
If the force of gravity is perpendicular to the surface where the ball meets the earth, that hard surface prevents the ball from going anywhere so we are saying that the ball can potentially gain no speed thus it has "no potential energy".  
This is where the confusion starts, because in fact that ball had potential energy (if only allowed to move). The only place where the ball could really have zero potential energy would be at the lowest point on the surface of the earh (or that of the surface of the ocean floor depending on the density of the ball)
If we move the ball to the side making available a cliff where the ball could potentially roll down on, we realise that the ball (or the system) did have potential energy, we just said it didn't, for practical reasons.
A: When An object is on a cliff at rest then relative to ground ,it will have 0 kinetic energy.we can take the surface of the cliff at 0 potential. When it is on surface of cliff,Two forces are working on it, The gravitational force of earth and the normal force of surface.Net force is zero, So it is in equilibrium.As net force is zero,so change in it's kinetic energy or its potential energy is also zero ($F=-dU/dx$).
When it is falling from the cliff ,we removed the normal force ,so net force is acting that is gravitational force. This will change its potential (kinetic)energy. We assumed the cliff at 0 potential energy ,so the force will decrease the potential energy in negative as wolframe johny said.
The “energy” is just an abstract accounting tool that we use to describe the results of those interactions in terms of the work being done.
We talk about the potential energy of system ,because we need force that will cause change in potential energy,but kinetic  energy is just because of the relative motion of the object
A: If the ball was on top of a cliff, then it already had gravitational potential energy. This comes from the fact that if the ball fell off the cliff, if would gain kinetic energy (while losing the potential energy it had from being on top of the cliff).
Gravitational potential energy is usually discussed with respect to a system because there needs to be other mass(es) exerting gravity on an object to give it said potential energy. However, there are other types of potential energy internal to an object (e.g., chemical potential energy), that has no reliance on an external system.
The main thing to remember is that potential energy is how much kinetic energy an object could potentially exert. An object being on top of a cliff has potential energy since it has the potential to drop off the cliff and gain kinetic energy.
