Radar is an acronym stemming from "RAdio Detection And Ranging." A ray of electromagnetic energy is beamed out from a transmitter/receiver called a radome. The beam is reflected off objects in its path, and the strength of the return signal is used to determine precipitation intensity. A second pulse sent a fraction of a second later is used to measure velocity toward or away from the radome, emulating the Doppler effect and turning radar into Doppler radar.
Dual-polarity refers to the two parts of the beam at a 90-degree angle from each other (imagine a vertical ray and a horizontal ray coming out from the radome). This is used to estimate the shape and size of the rain, snow or hail being observed. Originally, in the early 1940s, radar was used in aircraft to detect other aircraft, but the device did not work as well in precipitation because of all the clutter. Then the idea to use radar to detect the clutter was born.