The single cell voltage is 3.7v (nominal). The nominal voltage is the average working voltage of the circuits the cell is normally used in, the cell itself starts out fully-charged at 4.2v and is flat at 3.2v. To get a higher voltage you can use two cells in series, for double the voltage. To get, say, 5v from the resulting (average of) 7.4v you can either put in a resistor of fixed value, or variable (but it will need to take a lot of current so this is not practical), or an electronic circuit called a regulator, to drop the voltage. The advantage of a regulator is that it provides a constant voltage. How constant that is will depend on the quality of the circuit. The current supplied by the battery is that demanded by the atomizer, plus some to run the circuit, plus a tiny bit for overhead (what the circuitry loses in inefficiency).
So if the voltage is higher, it's easy to drop it. If the voltage you start with is lower, it's a bit more of a problem. To raise the voltage, you need a booster - an amplifier to get the 3.7v up to 4.5v or whatever you want. Nothing's for nothing, so to do this will draw a lot more current from the battery. It draws the atomizer current (~2 amps) plus some more amps to get the extra voltage. It uses the equivalent of a power amplifier or PA, exactly like your hifi does, but instead of increasing the sound volume it's increasing the voltage, using more current as the fuel. So now you have a booster regulator - an amplifier to provide a voltage (which can be variable) higher than the starting voltage, plus a regulator to keep the voltage constant.
If you need maybe lower voltage, maybe higher voltage, you need a special kind of regulator: a buck/boost regulator. The 'buck' is the voltage cut function, if needed, the 'boost' is the voltage amplifier function, and the regulator keeps it level. There are three circuits packaged into one: a dropper, an amplifier, and a leveler.
With DC like this it's all fairly simple* as you don't have to deal with conversions in either direction to/from AC, with the ripple and waveform and frequency issues that brings. PSUs drop AC to lower DC values and smooth out the sine wave of AC. They use very little current as it's an easy job.
*But the clever bit is getting it all in such a tiny package, and one that does not simply melt as a result of the power going through it. This is the part that took about 100 years to achieve.
Inverters jack up the low DC voltage to mains electricity levels and introduce a waveform to create AC. They use a tremendous amount of current to do the job - getting 12v up to 110v or 240v takes about 100 amps at the battery if you want a decent amount of power out of the other end. Cheap ones produce a nasty stepped square wave, not a pure sinewave, and their frequency is unstable (60Hz US or 50Hz in the EU). You can think of the boost part of a buck/boost regulator in an e-cig as a DC-to-DC inverter.
