Defining a volt, and seeing what that means in practice
The volt is defined in terms of energy shifting from store to store by flowing charged particles, here moving around a circuit.
1 volt = 1 watt1 ampere
You can understanding the implications of this by thinking about a simple circuit.
Set up a situation where current in the resistance (such as a bulb) is one ampere and the voltage across the bulb is also one ampere. As a result one joule of energy is being shifted per second. (This is electrical working; see SPT: Energy topic).
Set up a situation where the current in the resistance is smaller and the energy being shifted will also be reduced.
Accumulating energy in a store
So, for a fixed voltage, the energy accumulates steadily in the thermal store of the surroundings. The energy shifting from the chemical store also depletes that store by the same amount each second. Change the flow of charge, and you change the energy shifting: the power.
Changing the voltage
Arrange for the voltage across the bulb to be 3 volt. As a consequence the accumulated energy shifted to the surroundings as each coulomb of charge passes is 3 joule.
The voltage across the bulb (or any other resistor) determines how much energy is shifted to the surroundings as each coulomb of charge passes through. The voltage is the energy shifted per unit charge.
The voltage across the battery sets the amount of energy which is shifted from the battery per coulomb of charge that has passed. You can rewrite the definition above for a duration of one second:
1 volt = 1 joule1 coulomb
This focuses on the accumulated effect of the flow of electrical charge, rather than the process as it happens. As the brightness of bulbs depends on the power, rather than the accumulated energy, this may make better connections with what is seen and easily felt to be important.