What defines Volts, Amps, Watts and mAh?
I have stopped thinking about what the electrons do and what they don't and how they figure out which way to go and how they know that the left path has less resistance and go there more - so I just stick to things I can remember easily and which have some practical meaning to me:



Volt is the unit for energy per charge, that is joule per coulomb. The more volt you have the more energy a single charge has.
$$1~\text{V} = \frac{1~\text{J}}{1~\text{C}}$$



Ampere is the unit of charge flowing per time, so coulomb per second. The higher your amperage the more charges will flow in the same amount of time.
$$1~\text{A} = \frac{1~\text{C}}{1~\text{s}}$$



Watt is the unit for power, power is energy per time (always, doesn't matter if mechanical or electrical).
$$1~\text{W} = \frac{1~\text{J}}{1~\text{s}} = 1~\text{V}\times1~\text{A} (=\frac{1~\text{J}}{1~\text{C}}\times\frac{1~\text{C}}{1~\text{s}})$$



mAh is a unit of charge - why? m is milli, so just a SI-prefix meaning the number times 10^(-3). A is ampere, and h is hour and multiplied, so:
$$1~\text{mAh} = 1\times10^{-3}~\frac{\text{C}}{\text{s}}\times3600~\text{s} =3.6~\text{C}$$

Milliamperehours are typically used to give an indication about battery capacity. For practical applications, you usually would want the energy which is stored in the battery, so watthours would be a better measure. Amperehours are fine if you have two batteries of the same chemistry and configuration - otherwise it gets pretty useless. A 12&nbsp;V 1&nbsp;mAh battery has more energy than a 3&nbsp;V 3&nbsp;mAh battery.



And for completeness, because people always mix it up: watthours are a unit of energy because it is power times time and not power per time. So you have a device running at 1 W for an hour, that is one watthour.
$$1~\text{Wh} = 1~\frac{\text{J}}{\text{s}}\times3600~\text{s} =3.6~\text{J}$$