How can a battery 'come back to life'? Earlier today, one of my family drained our car's battery by accident, by leaving the lights on too long. It wouldn't start at all, so they had to walk back home from where the car was parked (a few streets away). However, when we went over there later on in the day, with our other car and some jumping leads, it just started up right away.
How would/could this happen? Why would the voltage/charge seem to 'recover' over a period of time, when previously it was drained?
 A: In general, any electrochemical reaction in a battery creates concentration gradients of different compounds between electrodes, which tend to form thin layers of different substances near them. Some of these layers, if solid, precipitate on the electrodes. Depending of battery's chemistry and charge/discharge mode this can result in an insulating film that would interfere with the battery operation. This is, for example, the reason why $\text{NiCd}$ batteries have the "memory effect", and all (IIRC) batteries based on lithium chemistry can stop charging completely and "play possum", but reverse under tens of hours of steady low charging current due to an eventual dissolution of the parasitic insulating layer.
In the case of lead-acid battery, discharge involves the formation of solid $\text{PbSO}_4$ on both plates. Normally this sulfate is spongy, porous amorphous solid that lets the acidic electrolyte through to reach metal and metal-oxide electrodes under it, and also dissolves easily when a charge current is applied, due to its large cumulative surface area. But under certain conditions, such as constant low discharge current and low battery temperature, the sulfate layer may grow in a densely packed crystalline form, and produce an insulating layer that prevents further discharge. This layer, while it is still thin enough, may eventually let some electrolyte to soak through spots when the load is off. When you connect a low impedance load (starter) to the battery, the thin brittle insulating layer may break down starting from these pores where electrolyte is now present, where it experiences mechanical stress due to thermal expansion, allowing the discharge reaction to continue.
In extreme cases, when the battery spends long periods of time in a deeply discharged state, the amorphous sulfate on electrodes transitions into a densely packed microcrystalline form, resulting in a solid white dense impenetrable layer that no longer dissolves when charging is attempted. It forms primarily on the negative plate of the battery (I just realized do not know why, really, as you got me thinking about it! Furute research required...). This is known as sulfation, and is generally irreversible. This is also one of the processes that limits lifetime of the battery even under optimal use (positive plate corrosion being another one).
