Is battery mAh measured per cell

As far as battery technology for a laptop, what you've got is what you're stuck with for the most part as the charging electronics to different battery technologies are internal to the laptop itself.

That said, you've asked good questions about the terms used to identify batteries - mAh (yes, milliamp hours) and W (watts).

Watts are a standard way to measure power output of any electrical device. The best measure for a battery would be in Watt-hours, measuring actual energy capacity of the battery. (Watts = Volts * Amps)

A second-best measure would be the mAh measure which would indicate how much current the battery can supply over time at the specified voltage. (I used this figure frequently when selecting AA rechargeable batteries.)

I've not heard about voltage variations between batteries. Essentially the most efficient battery will be the one with the voltage that best matches what the computer's internal electronics were designed around. Any excess will be frequently lost as heat as the internal electronics convert the voltage to the proper level (which is probably not a figure specified by the manufacturer - they're more likely to supply a figure denoting the range of voltages the laptop will accept).

Bottom line for most batteries is that shelf-life is still a factor, so look for a reputable dealer with an indication of when the battery was manufactured since an older better specified battery may not have the capacity of a new battery with poorer specifications.

It sounds like you are talking about the kind of thing that has USB power ports for charging cell phones and tablets. If that is the case, then no, you can't power a laptop with it. What you want is a UPS.

Also their power rating is usually expressed in terms of a single cell, so 16 Ah at 4 volts = 64 Wh. Laptop batteries normally tell the capacity of an individual cell and how many cells they have, so Your 6 cell 4400 mAh laptop battery holds 105.6 Wh.

A typical UPS has a 7 or 8 Ah 12V battery so it holds 84 Wh. Higher end ones have dual batteries for twice the capacity, and output mains power for your laptop charger to convert to whatever the laptop accepts.