Piezo electric water vaporizer
The device is called a nebulizer or atomizer. 

It's just generally just a self-resonant Colpitts oscillator with a single power transistor and some passives (including a couple inductors). There may be some additional circuitry to turn it off if the water level drops. 

I suggest you study the prior art in US patent 4996502 assigned to Japan's TDK Corporation.

 

The droplet size is inversely related to the operating frequency (also the mechanical resonant mode of the piezoelectric element). Typically for consumer units a frequency around 1.5-1.7MHz is used, which generates droplets of a few microns in diameter.