In general, the different size of cell batteries are to increase the amount of power they have

like both a AAA and a D cell are 1.5V, but they're radically different sizes. is this for increased amount of power the larger the battery you get? so they last longer?

Battery size is mostly driven by the amount of energy stored. This is especially true of the standard retail cell types, like the 1.5V AAA's vs. 1.5V D-cells. They provide the same voltage, but the larger D-cell holds more charge, so it takes more time to exhaust.

You can test this by wiring together each battery type to a little flashlight bulb: same voltage = same brightness. And if you're really patient, you can wire up two such rigs, and watch to see the D-cell outlast the AAA.

To a lesser extent, size can reflect voltage, in that batteries with higher voltage require more internal cells stacked in series. Theoretically, such stacks can be any size, including very small sizes. But in practice, higher voltages are just easier to make in bigger packages.

Voltage comes from the difference in the electric potential of the internal chemicals: simplistically, one wants to give up an electron, the other wants to accept it. The sum of these two gives the voltage of the battery. To get more voltage you have to stack multiple chemical cells in series.

Yep, they are able to put out more amp hours. 8 AA batteries in series is 12v but there is a reason you cant use that jump your car if you needed to :). Generally, the larger the battery, the larger its capacity and the more current can pass through at a given time.

yes