aside: The water pipe analogy is actually quite accurate, but most high school students don't learn fluid dynamics and the heavy calculus makes it more trouble than it's worth. (If you treat volumetric flow rate as akin to current, and pressure as electric flux density, then using Bernoulli's equation neglecting gravity, assuming constant area and velocity along a pipe/wire, it all works out).
We have a device that can measure "voltage" between two points (not just in a circuit but in the air, on the ground, wherever you place the terminals).
The first thing is that if we measure the "voltage" between A and B, and then between A and C, the voltage between B and C will be the difference between these two—if you freeze time, at least. So this means you can pick whatever reference point you like, then any other point in the universe has a certain electric potential (measured in volts) relative to the reference point. So, there is something in the fabric of space that can be "stronger" or "weaker" than other parts of space. This is what we call the "electric field".
Note that this is not about electrons. It is possible to have a positively charged plate and a negatively charged plate and a total vacuum in between, but this vacuum will linearly transition from one potential to another. (I.e. the voltage across the left half equals the voltage across the right half.)
In a simple electric circuit, each point has a fixed voltage (usually measured relative to "ground"). Any loop through the circuit will involve exactly the same amount of "going up" as "going down".
current and resistance
Current and resistance are probably more intuitive in the context of electric circuits. Current is roughly speaking a count of how many electrons pass through an area (e.g. the cross-section area of a wire or resistor) per unit time. Resistance is how many volts will produce 1 amp of current flowing "across a resistor". Technically this means measuring the voltage between both ends of the resistor, and the current at some cross-section anywhere before, in, or after the resistor. The current will be effectively the same anywhere you you section (not beyond branches in wires of course).
(Obviously we also have devices that can measure current and resistance; I made a point about measuring voltage because electric fields are the hardest thing to grasp.)
more on current and resistance
When electrons are free to flow, they are driven to equalise electric fields. The stronger the field, the more electrons will flow. However, different materials will resist electron flow to different degrees, hence differences in resistance (or rather resistivity). And electrons can be forced to flow the other way—batteries use chemical reactions, and generators use electromagnetism.
One way to see how current is proportional to surface area is this: assume you have 1A traveling along a wire, and you have 1A travelling down another wire, that's 2 coulombs per second. If you squash the two wires into one, the surface area will double and you will still have the same number of coulombs per second — 2.