A “Geiger counter” is named for the Geiger-Mueller tube, which is the core component of this device.
This tube contains a thin metal wire through its center. The space around the wire is sealed and filled with gas. The wire carries a charge of 500 volts relative to the tube. An atomic particle or gamma quant penetrating the tube (or an electron knocked out of the wall by X-rays or gamma rays) tears electrons out of atoms in the gas. Because of its high positive voltage, the wire attracts those electrons, which gain energy in the process. They collide with atoms, releasing more electrons, until the process cascades into a “waterfall” which produces an easily detectable pulse of current.