Physicist: Chemical batteries use a pair of chemical reactions to move charges from one terminal to the other with a fixed voltage, usually 1.5 volts for most batteries you can buy in the store (although there are other kinds of batteries). The chemicals in a battery litterally strip charge away from one terminal and deposite charge on the other. In general, the more surface area the chemicals have to deposit charge onto, and take charge away from, the higher the current the battery can produce.
The best way to represent the way a real battery works is to replace the battery in a circuit with an ideal voltage source (which is what we usually think of batteries as being) and an imaginary resistor called the battery’s “internal resistance“. The internal resistance can be used to describe why an AA battery is incapable of generating an arbitrary amount of power; the more current that the battery creates, the more the voltage across the internal resistor drops according to Ohm’s_law (V=IR). You can picture this as being a little like pushing a cart; if the cart isn’t moving you can really put your shoulder into it, but as the cart moves faster it becomes harder and harder to apply force to it.
A car battery only produces 12 volts, which is the same as 8 ordinary batteries in series. That voltage is so low that you can put your dry hands on the terminals of a car battery and feel nothing (please don’t trust me enough to try it; I don’t even trust me enough to try it). And yet the internal resistance is so low that if you connected the terminals with a normal wire, the current in the wire would be so high that the wire would melt or explode.
You can model the way a battery dies by increasing the internal resistance. A nearly dead battery still provides 1.5 volts, but has a very high internal resistance so that drawing even a trickle of current zeros out the voltage gain.
The voltage across a capacitor on the other hand is always proportional to the charge presently stored in the capacitor (this is the defiition of capacitance). You can think of a battery as being like a water pump, always providing the same pressure, and a capacitor as being like a water ballon, the pressure increasing the more water is in the ballon. The amount of energy in a capacitor is much easier to measure because of this (if you can measure the voltage across it, you can know the energy immediately). But, because the voltage supplied by a capacitor changes dramatically as it drains, special adaptive circuits are needed to step down the voltage to a fixed, consistent level in order to power a device. Alternatively, the device can be made to work over a wide range of voltages, but that tends to be more difficult.
It’s only fairly recently that capacitors have become small and powerful enough to store energy on par with chemical batteries. A few decades ago electrical engineers would prank the new guy by asking them to go into the backroom for a 1 Farad capacitor, which at the time was ludicrous. The poor sap would be back there forever (electrical engineers think they’re so funny). However, that joke has run its course, because today you can buy an over-the-counter, several-thousand Farad capacitor, that’s small enough to fit in your hand (and they pack a punch)!
So, as a general rule of thumb, batteries have a fixed voltage but:
big or new batteries tend to have a low internal resistance, so they can deliver a high current
small or old batteries tend to have a high internal resistance, so they can’t deliver much current