What does "28 volts DC" mean, really? The common battery voltage on aircraft is commonly referred to as "28 volts".   What does that really mean in terms of a lead-acid battery? 

In many uses, such as cars ("12V"), trucks ("24V") forklifts and golf carts ("36V" or "48V"), locomotives ("64V"), the nameplate number is exactly twice the number of lead-acid "cells".   But in reality, the fully charged/charging/float voltage of the system with alternator running is higher - cars 14V, trucks 28V, locos 75V.  

Battery units are generally sized at 3, 6 or sometimes 4 units (6, 12 or 8 volts nominal). 

So, when aircraft power is specced at 28 volts -- do they actually  mean a 14-cell lead-acid battery or  two 7-cells (which is almost unheard of)... or do they actually mean a 12-cell battery with a "float" voltage around 28V, (built out of two common-as-dirt "12V to everyone else") batteries? 

I hope this isn't overloading this question too much, but when nickel-cadmium packs are used, does that have 20 cells, or 19 or some other number?
It's the second option. In any other context, a small aircraft's electrical system would be called "24V". But the alternator regularly puts out 28V, so that's what's typically expected if you hook up your multimeter while the engine's running. Of course, the on-board equipment can operate over a wide range of voltages, often down to as little as 20V.