Terminal voltage of DC battery
Let's get the easy one out of the way. AC batteries can never, and will never happen, because the chemical reaction that creates the voltage will only ever take electrons from one electrode and deposit them on the opposing one. To have AC instead of DC, we can use a device called an inverter to switch the DC back and forth at the correct frequency, and do some other cool things to make it a better source of AC power than just an ordinary square wave. 

Now the terminal voltage of a battery... 



Here is a simplified model of what is going on. When you pick up an unknown battery, you have no idea what it's internal resistance is; if the battery is really old or damaged, it could be very high. And since real voltage sources are nothing like the ideal one is this model, you do not know how much instantaneous power it is capable of outputting. 

So let's say this battery is outputting exactly 12 volts, but it's internal resistance is really high. If you put an ordinary voltmeter across it's terminals, that voltmeter doesn't draw very much current, so it will register 12 volts. When you attempt to use the battery, the current across the resistance reduces the voltage at A significantly, and you can now see that the battery is not usable.

The same thing can happen to a battery that has very little charge left, even when it has normal internal resistance. When you attempt to draw a load on the battery, the voltage source simply falls in value, as the chemistry no longer has enough energy left to sustain it. In this case, it is typically more obvious, as the voltage will be abnormally low when read by a voltmeter, but the fact remains that it will show a voltage, and that voltage might look adequate, especially in a lead acid battery. 

There are professional grade handheld testing units that can test a battery by forcing a very brief high current discharge and concurrently measuring voltage and current. These readings then allow you to directly measure internal resistance, and can also be used to estimate a value known in some industries as Cold Cranking Amps. You can use a similar method on a smaller scale to verify that a battery has at least some usable capacity left by putting a reasonable resistance in series with the battery and taking voltage across and current through it.