The reason it is called "precipitable" water vapor (PWV) is that the water vapor is in the form of gaseous vapor which may further condense into cloud form and then into actual precipitation i.e. rain (which often overwhelm the sensors). It is often also called total column water vapor which more accurately reflects what is being measured i.e. a height of the equivalent column of liquid water, hence the measurement in mm. The American Meteorological Society glossary entry on precipitable water (or precipitable water vapor) similarly defines it as:
The total atmospheric water vapor contained in a vertical column of unit cross-sectional area extending between any two specified levels, commonly expressed in terms of the height to which that water substance would stand if completely condensed and collected in a vessel of the same unit cross section.
The total precipitable water is that contained in a column of unit cross section extending all of the way from the earth's surface to the "top" of the atmosphere.
For very high and dry sites e.g. the Atacama, Antarctica will have less than 5mm PWV, sea level sites will be 50+mm of PWV. ESO's Cerro Paranel observing site (home of the VLT), has median PWV of 2.5mm (see e.g. the histograms in this ASM 2016-2018 report). The Atacama Pathfinder EXperiment (APEX) sub-mm telescope on Chajnantor (also home to the ALMA array) has had a weather station for several years and for example the APEX 2017 weather statistics show that the APEX site (~5100m altitude) had a PWV <1.5mm approx. 67% of the time.
There are several ways to measure the PWV. The Earth orbiting satellites such as the AIRS and MODIS instruments on NASA's Aqua and Terra satellites measure the radiance in several IR wavebands (typically from approx. 0.5 to 15 um). Some of these bands are essentially dominated by water absorption bands so performing differences between bands containing water and without water will give a measure of the water column.
Microwave radiometers at 210 or 225 GHz are traditionally what have been used to measure PWV, with more modern radiometers shifting to the 350um (856 GHz) window/band. An example is the one at the Caltech Submillimeter Observatory on Mauna Kea described in this link at Gemini Observatory. They work similarly by measuring the depth of the absorption by the water vapor in the molecular lines/bands. More water vapor produces deeper lines with more optical depth. More details are in this (freely available) paper by Radford which shows the effect of increasing PWV on the absorption in the sub-mm, the amount of PWV at various sites and more info on the measurement.
Finally this can also be done by dual frequency GPS receivers at the L1 (1575 MHz) and L2 (1224 MHz) and measuring the path length excess through the atmosphere above what is caused by the receiver-satellite distance alone. By using two frequencies, you can remove the effect of the variable ionosphere on the path length. The remaining zenith path distance breaks down into two parts, a hydrostatic or "dry" component which can be easily estimated using e.g. the Saastamoinen model, leaving the "wet" component caused by water vapour. Once this is corrected to the zenith, this gives a measure of the PWV.