I'm looking for an explanation of voltage that doesn't compare electricity to water and voltage to pressure?

Voltage is a measure of the amount of energy that each electron has. More specifically, it is the difference in that energy between two points in a circuit.

The unit of EMF is the "volt", which is defined as one Joule of energy per Coulomb of electrons.

A 9V battery gives 9 Joules of energy to every Coulomb that passes through it, and pushes them toward the negative terminal. A resistor that has 5V being dropped across it means that each Coulomb that passes through it loses 5 Joules of energy (usually to heat).

In layman's terms, voltage is a measure of how much "kick" each electron has when going through the circuit. Higher voltages mean that each electron is much more lively and energetic, and can do more work, than lower voltage ones do.

It's in this sense that voltage is compared to pressure. Voltage is the force that compels electrons to move through a circuit, and pressure is the force that compels water to move through a pipe.

I = V/R; where I is current, V is voltage, and R is resistance.

a = F/m; where a is acceleration, F is force, and m is mass.

And there you are. Think of voltage as the force (not pressure) that pushes those electrons along against some resistance to their moving. mass m is like the resistance R in that mass represents inertia which is a resistance to change in velocity, which is acceleration. And if you think of force as a push or a pull, then voltage is a push or pull on electrons.

In physics lab, we used to use I = V/R on analog computers to model a = F/m. So for a given V, say, 10 v that would represent say 10 Newtons of force. And R in ohms would be varied with 1 ohm per kg. The result in amperes would represent the acceleration which would graph out against the varying ohm as mass.

I think of voltage as height, and the analogy works extremely well. That is because the form of the gravitational force F = Gm1 m2/r^2 and the electrical force F = k q1 q2/r^2 are very similar.

So long as you remember that unlike gravity, in electricity there are negative charges (things that like to roll uphill), that works fine.

I know why they use the water pressure analogy. It's so that you have an analogy for resistors in terms of narrowness of pipes. Put resistors in parallel and the pipe is wider. There's less resistance.

I don't have a good resistor analogy for when I think in terms of height and balls rolling downhill. Unless you want to keep thinking about a pipe, but this time a pipe full of junk, so there is "resistance" as the ball rolls downhill through the pipe. Wider pipe, less resistance. That is in fact what happens to electrons inside resistors.

voltage is a measure of potential energy per unit of charge

if it takes 1 J of work to move 1 C of charge from one place to another

then

the 1 C of charge has acquired 1 volt of potential in the move

and

the two places have a potential difference of 1 volt

a volt being defined as a Joule per Coulomb

take gravitational potential as an analogy

if a table is 1 m high and a chair is 0.5 m to the seat

then moving a mass of 1 kg from the seat to the table top takes mgh work

or 1(9.8)(0.5) = 4.9 J

so the change in potential is 4.9 J

now

since we raised 1 kg we could talk about the potential difference between the seat and the table in terms of energy per kg

in this case 4.9 J / 1 kg or 4.9 J/kg

we could now use this ratio to find out how much work it would take to lift 4.5 kg from the seat to the table

4.5 kg * 4.9 J/kg = 22.05 J

thinking of volts as a measure of 'energy density' may help

a potential difference of 1 million volts is scary and rightly so

but

if only 1 millionth of a Coulomb moves across this potential difference

then only 1 J of work is done - not a big deal

however

if just 1 Coulomb moves across

then 1 million J of work is done - really big deal

hope I didn't just add to your confusion