Dry bulb is a normal temperature reading with say a thermometer. Wet bulb is that same thermometer but it is wrapped in a wet cloth to simulate evaporation of sweat.

The purpose of wet bulb temperature measurement is to fix the dangerous temperature threshold at body temperature instead of having to adjust for humidity. So if the wet bulb temperature crosses 35C/95F you know that it is dangerous to even be outside because your sweat can't even evaporate enough to prevent you from overheating just standing in the shade.

Dry bulb is the temperature independent of humidity. Wet bulb is has a wet cloth on the thermometer bulb. This simulates how much sweat cools you in the current humidity and wind.

Measuring humidity instead and cross-referencing to get heat index is more common these days, but IMO it's worse. 120 in the desert vs 120 heat index due to humidity is the difference between someone using a hair dryer on your face and getting cooked in a steam room, and it doesn't consider wind and cloud cover.