Understanding voltage, what is physically happening with the electrons? The way I understand voltage vs current is that voltage is the potential energy (or force) of the current. And the current is the amount of charged electrons passes through a conductor at a point every second. I think of voltage as the pushing force of the electrons in a conductor.
If this is a correct way of looking at it, what actually is that force of voltage? Technically force$=m*a$ but is speed a factor? Do the electrons actually move faster in a 220V system vs 110V, or for that matter for higher voltage in a DC system? And what is the source of this force?
 A: 
The way I understand voltage vs current is that voltage is the
potential energy (or force) of the current.

The voltage $V$, or potential difference, between two points is the work required per unit charge to move the charge between the two points. It is the potential energy per unit charge, not the total potential energy or force of the current.

And the current is the amount of charged electrons passes through a
conductor at a point every second.

The electric current through a surface is defined as the rate of charge transport (positive or negative charge) through that surface.

I think of voltage as the pushing force of the electrons in a
conductor.

The electric field provides the pushing force for the electrons in a conductor. The electric field is the gradient of the voltage (volts/meter).

If this is a correct way of looking at it, what actually is that force
of voltage?

The force is supplied by the electric field. For a constant electric field $\vec E$ the force applied to a charge $Q$ is $\vec F=Q\vec E$. Then, since the electric field is the gradient of the voltage over the distance $d$, the magnitude of the force on the charge is $F=\frac{QV}{d}$.

Technically force$=m*a$ but is speed a factor? Do the electrons
actually move faster in a 220V system vs 110V, or for that matter for
higher voltage in a DC system?

It depends on what you mean by "is speed a factor".
Let's assume your "system" is a resistor of resistance $R$ connected to an ideal 220V or 110V voltage source.  Then, per Ohms law, the current with the 220V source will be twice that for the 110V source. For the same resistor, the electron drift velocity (the average speed of the electrons going in the same direction opposite the field) for the 220V source will be twice the electron drift velocity for the 110V source.

And what is the source of this force?

The source of the force is the electric field established by the voltage source.
Hope this helps.
A: Electrons are accelerated by electric fields. But the story of what happens in a wire is a little bit more complicated.
Electrons have thermal energy, and the free electrons in a conductor are moving randomly quite rapidly. At room temperature, electrons are moving somewhere on the order of 10^5 meters/sec. By comparison, the speed of light is about 3 x 10^8 meters/sec, while the drift velocity of electrons (which will be explained soon) in a typical circuit might be on the order of 10^-4 meters/sec.
Now, thermal energy is associated with random motion. So, although the speed of electrons at room temperature might be around 10^5 meters/sec, the average velocity of electrons when there is no electric field, and no electric current is zero. That is, because the motion is random, the probability that an electron is moving with velocity $v$ is the same as the probability that it is moving with velocity $-v$. So, when averages are taken, the average velocity (which is a vector quantity) is 0.
When an electric field is applied to a conductor, that electric field will cause electrons to accelerate in the direction of the electric field. All of the free electrons, regardless of their initial velocity (speed and direction) will gain some velocity in the direction of the electric field. This will make the average velocity, no longer zero. This average velocity of the free electrons in a conductor is called the drift velocity, and it is quite small. Again, something on the order of 10^-4 meters/second is in the range of typical.
If the conductor were a superconductor, and the electric field were continuously applied to the electrons, the electrons would continue to accelerate (up to a point). However in non-superconductors, electrons "collide" with with the atoms or ions in the conductor very frequently. It is not important whether these "collisions" are "collisions" in the sense of what we are familiar with in macroscopic bodies. The essential fact about these collisions is that they randomize the velocity of the electrons, so that the average motion of electrons just after collision is again zero. (Incidentally, the randomization of current into thermal motion is what makes conductors (except super-conductors) dissipate energy. From a macroscopic point of view, this process is called Joule heating.).
How far do electrons accelerate before they collide and have their velocity randomized? In room temperature copper, the distance is only about 4 x 10^-8 meters. It is because electrons have such a little distance to accelerate before their velocity is randomized, that the average velocity, (i.e. the drift velocity) is so low (i.e. 10^-4 meters/second typical).
A: 
I think of voltage as the pushing force of the electrons in a conductor.

That's not a bad way to think of it.  No voltage, no (bulk) motion.

force=m*a

That's true for net force and net acceleration.  But at the scale of a single electron, (the charge carrier we deal with most often), there are lots of other competing forces.  So we don't expect the force we apply to directly turn into an acceleration.  And dealing with the mass of the charge carriers is inconvenient as well.  So just avoiding this entirely is usually best.

Do the electrons actually move faster in a 220v system vs 110v,

Forces don't directly relate to speed and neither does voltage.  In a purely resistive circuit where all the resistances were identical, then the current would be higher and the drift speed would be higher for 220 vs 110.
But most circuits aren't like that.  If you plugged your laptop into a 220 vs a 110, the 220 circuit would supply the demand with less current and the drift speed in the 220 portion would be lower than the 110 circuit.
Think of it like asking if the chain on your bicycle moves faster if you push on the pedals harder.  If you stay in the same gear, and you don't change if you're going uphill or downhill, then yes we'd expect it to move a bit faster.  But if you're pushing harder because you're going uphill, then it might go slower.  Or if you switched to a low gear, the chain might go slower.
A: Without getting to into the math Potential energy is all about energy/work.  Electric fields are about force per unit charge
If there is a potential difference from point A to point B of $V_{0}$ volts then this is the amount of work that the field would do on an object per unit charge, if it were to move from B to A.
Thus, If  there is a potential difference of -220V from point A to B, then the total amount of energy gained by a unit charge as it moves through A -> B would be greater than if there was a potential difference of -110V across A to B
This is CLOSELY related to the "Push" that electrons feel in a potential difference, in that the "summation" of all the "pushing", is greater for higher potential differences. This is because more work is done on particles moving through  higher PD's. 1/2mv^2 = -V  Meaning a higher FINAL speed once it has traversed A to B
This doesn't necessarily mean that the electric field for higher potential differences is greater.
In the simplest example,
V = ED
If there is a potential difference of $V_{0} $volts across a 1m gap,
this would have a higher electric field then if there was the same potential difference across a 2m gap.
The final kinetic energy of particles traversing the Pd will be the same , But for  the 1m situation, particles experience a higher force, for a shorter time.
