How does this electrostatic generator operate?
The piezo igniter generates high voltage.  That's its job.  

The high voltage normally used to make a spark to ignite the gas in a cigarette lighter.

The piezo igniter generates AC, though.

By sending it through a diode, he's converting the AC to DC.

Which polarity he gets is determined by which way the diodes are pointed.

He uses two sets of diodes so that he can switch polarity 

Regardless of which way the diodes are pointed, he gets charged up to the opposite polarity of the free end of the diode set - the user is part of the circuit.  If the diodes are pointed such that the free end is positive, then the user will end up with a negative charge.

Air has a breakdown voltage of around 3000 volts per millimeter.  The beginning of the video shows the igniter making sparks of around 3 or 4 mm.  That's somewhere around 9000 volts needed to jump the gap.

Diodes rated to withstand voltage that high aren't as common as diodes with lower voltage ratings.  That's why he's using a bunch of them in series.  He has diodes rated for lower voltage.  Putting them in series adds the voltage ratings.  If they are each rated for 1000 volts, then eight of them in series will withstand 8000 volts. (More or less.  True enough for playing around, not quite true if you need to build a reliable high voltage device.)

The diodes play no part in generating the high voltage.  The igniter does that.  The diodes are just to make the generated voltage "point in the right direction."



You haven't said what you want to do with the high voltage.

If you just want to play around a little bit, then the method shown in the video will get you a (relatively) safe high voltage for experiments.  "Safe" in the sense that the current is very limited and therefore unlikely to injure or kill you.

If you need a constant source of high voltage but low current, then you could look into purchasing a high voltage DC power supply.  They can be bought with output voltages in the 10s of thousands of volts and current in the milliampere range.

If you want to build such a high voltage supply, then you will want to look into cockcroft walton voltage multipliers.  They are fed with AC, and generate multiples of the AC voltage as DC.  You generally want to start with a fairly high voltage, and multiply it a few times.  You don't want to start with 5VAC and try to multiply it to 10000VDC - that'd take a lot of diodes and not be very efficient.

If you are really just interested in electrostatics, you can make high voltages without much in the way of electronics.

There's Van de Graaf generators that can make high voltages.  They are usually motor driven.

Then there's Wimshurst machines.  Small desktop machines of that sort are generally hand cranked.  The one I built reaches about 30000 volts with just a few cranks of the handle.  It does make DC, but the polarity is not fixed unless you do something to force a particular polarity.  This "special step" usually consists of mounting a small ball of wool in a particular spot.

You can also get parts made for old color TVs that used a cathode ray tube.  These had a transformer and voltage multiplier combination that they used to generate the high voltage needed for the picture tube to work.

So, really, there's a lot of ways to make high voltage DC. Which way you choose depends on what you are going to do with it, and how much power you need.