What is the physical meaning of electric potential, potential difference, and voltage? When resembling the electricity flow through a wire to people walking through a street: electrons are people, current is the number of people, resistance is the barriers on the way.
But what is the electric potential in this example?
What is the physical meaning of 2 V potential difference? And what is changed in the flow of electrons when the potential is 1 or 2 V?
 A: I'll give this a shot though I'm uncertain if this will clarify or cloud the issue.

electrons are people,

OK

current is the number of people,

No, that's not the correct analogy.  Current is a flow and a people (electron) current is a flow of people (electrons) and the amount of current is the number (Coulombs) of people (electrons) passing a fixed point in the street (conductor) per second.

resistance is the barriers on the way.

OK
It pays to keep in mind that electrons repel each other.  So, to improve this analogy, we need to have the people repelling each other.
To keep this simple, let's have the people walk single file and assume a person is attached to a spring that is attached to the person in front and also attached to a spring that is attached to the person in back.
So, if everyone is spaced equally apart and moving uniformly, there is no net force on any one person.
However, if there is resistance (barriers, friction, etc.), a person may slow down due to impact with a barrier and then feel a force from the spring that is compressed by the person behind and spring that is extended by the person in front.
It's not hard to see, extrapolating from this, that as the line of people move down the street, the springs are compressed more between the people at the beginning of the street than at the end of the street.
This means that the density of people is greater there; the people are closer together at the beginning that at the end.
So, the potential energy stored in the springs is larger at the end where the people enter than at the end where the people exit.  This must be the case since energy is lost to the barriers and friction as the people move down the street.  
Keep in mind that the flow of people is steady even though the density of people varies from greater at the beginning to smaller at the end of the street.
Similarly, when there is a constant current through a resistor, the charge density is greater at one end that at the other even though the flow of charge through the resistor is constant.
It is this charge density difference that gives rise to the voltage (potential difference) across the resistor.
A: 
But what is potential in this example?

In your people-in-the-street analogy, the electric potential would be a shop with sale in one end of the street, or maybe an accident or something else interesting. People tend to move towards the interesting end (less potential) and away from the other and less interesting end (higher potential). Just like a stone wishes to fall down and not up, because the points up and down are at different potential.

What is the physical meaning of 2 V potential difference?

The unit Volts is Joules/Coulomb, $[\mathrm{V}]=[\frac{\mathrm{J}}{\mathrm{C}}]$. Remember that $\mathrm{C}$ is the charge. So we are talking about energy per charge. And a higher potential has more energy per charge.
Now, back to the people-in-the-street analogy. If suddenly an accident happens, then people suddenly start moving towards it, because it is very interesting (less potential). Maybe they will run to get there fast. The closer they get, the slower they will move since they are soon there and they can already see more and more.
If you are a police officer or store security guard or whoever else holding people back, then the further away they are from the interesting spot, the more they wish to move - the more they push or the more they run. And here it is very hard to hold them back.
If you much closer to the interesting point on the street, then people can already see what is happening and they push less.
Like at a rock concert, if the speakers are bad, the people furthers away will push on the crowd to get nearer. As they come closer, they can hear the music better and better and they gradually stop pushing.
It's like their energy, with which they try to move, gets lower and lower as they get closer. The energy per person (meaning, the energy per charge) decreases as they reach the lower potential.
Does that make sence?

And what is changed in the flow of electrons when the potential is 1 or 2 V?

$1$ or $2 \mathrm{V}$ is just a measure of how much the potential is at some point on the street. If the usual situation (the street at all points before the accident happens) is $0 \mathrm{V}$, then when the the accident takes place suddenly you have a potential difference that will start dragging people in that direction.
If two accidents happen at the same time at different spots on the street, then maybe one involves a biker falling over ($1 \mathrm{V}$) and another involves a big truck with a tank of liquid nitrogen bursting, while a car catches fire and people are screaming ($2 \mathrm{V}$, the point is that is is more interesting).
They are not equally interesting, so the difference in potential $2 \mathrm{V}-1\mathrm{V}=1\mathrm{V}$ is towards the more interesting happening. If the lower potential accident didn't happen, then the flow of people would be more intens and faster, but since two events happened people are not sure where they wanna go. They do though see that the bigger accident is more interesting.
If two tanks of liquid nitrogen blows up at the same time at different spots (meaning, two points with the same potential) then some poor fellow who happens to be right in the middle, he is so confused that he doesn't know where to go! So he just stands there and changes his mind over and over and over, and never moves anywhere!
This analogy appears to be a society of indecisive people...
A: Since your question is kind of pictorial, i will try to stick with this and avoid any deeper mathematical / physical explanations. 

electrons are people

ok with that

current is the number of people

no, that's a mistake. Actually current is is this case the number of people crossing a line on the street per time. Like Steeven already mentioned in a comment 2 Ampere (let's assume 1 A is 1 guy to keep it simple) means that 2 guys more cross the line from left to right than the over way. How many people cross the line on the street in a direction is not represented by 2 Ampere.

resistance is the barriers on the way.

I am ok with that.

But what is potential in this example? 

I would rather compare it to a potential that is clearer: a hilly area. Electrical potential is similar to potential in gravity. So in that people example the electrical potential is the road going downhill. For people (electrons) going downhill no no energy is needed, they even lose potential energy. People staying at the same level at the street are gaining and losing nothing. People going uphill need energy to gain potential energy (for electrons this could be a magnetic field accelerating the electrons). Taking this explanation for electrical potential your next question is easy to answer.

What is the physical meaning of 2 V potential difference? 

It's just the gradient of the hilly road. If its kind of steepish you have a high Voltage and if it's flat you just have a small Voltage.

And what is changed in the flow of electrons when the potential is 1 or 2 V?

Remember my explanation of current? Raising the Voltage from 1 V to 2 V (making the road more steeply) has the effect on the electrons that more electrons per time cross the line from left to right than before. You can imagine this effect like, the road becomes so steeply that the people start sliding down (not everybody but more and more), if they want or not.
I hope this equation less explanation fits your needs.
Edit: to Rob Jeffries, i cant write a comment because i don't have 50 reputation so i am forced to answer this way.
I disagree with you that the analogy falls down because of a flat street.
Electrons in a conductor can move around too without any electrical field applied. As long as there is no preferred orientation (and people in a world with just an empty flat road have no preferred orientation) there is no current, at least no global current. It not just only don't fall apart it even perfectly fits into the analogy. Imagine a road first goes downhill, than is perfect flat and than goes downhill again. People gain velocity the first bit downhill and in an perfect world with only one road where are no losses (friction or what so ever) they keep the momentum and speed in the flat bit. They keep walking and walk faster when they reach the downhill part again. And here you have an people-electron explanation for electron drift in semi conduction structures. 
A: In my opinion electrostatic potential at any point in an electric field can be viewed as 
Suppose a positive charge makes an electric field
In that Field if any positive or negative test charge is present
Let us assume positive test charge
Then this positive test charge will tend to move away from Source charge due to force of repulsion 
And if it does not move. Definitely it should have enough energy available with it to counter this force of repulsion otherwise it would have moved away.
So Electric potential at any point in an Electric field may be defined as energy required to be possessed by a unit charge to hold its position countering the force due to E on it.
