A Geiger counter (or Geiger–Müller counter) is an instrument that is used for the measure of radiation. It was invented by Hans Geiger, who is perhaps more famous among physicists (though not the general public) for the Geiger-Marsden experiment, which was the experiment that led Ernest Rutherford to deduce the existence of the atomic nucleus.
Ionizing radiation such as from the three common forms of radioactivity: alpha decay, beta decay, and gamma decay, acts by ionizing normal atoms, that is, ejecting an electron from an atom. The radiation is detected by observing the electrical effects of this. Directly detecting an electric charge of a single elementary charge (one electron) is extraordinarily difficult even in modern times, and was out of the question in the first few decades of the 20th century.
There are two effects that can enhance this. First, a sufficiently energetic particle can ionize many atoms, making the electrical current much larger. For example, the alpha particle from the disintegration of a Radium atom ionizes about a million atoms of air before it comes to a stop. If the air is placed in an electric field connected to a sensitive charge detector, this can be detected. That was good enough for the early experiments of Marie and Pierre Curie—the cumulative effect of thousands of Radium disintegrations (that is, billions of elementary charges) could be detected collectively.
But this method could not "see" individual disintegrations. In order to do that, the ionization from a single atom needs to amplified into an electric current that can be detected electronically. A Geiger counter does this by turning a current of a million or so elementary charges into a much larger current; large enough for vacuum tube amplifiers to detect and turn into and audible "click" on a speaker or headphones.
A Geiger counter consists of an cylindrical metal tube, the Cathode, with an wire, the Anode, in it. The wire is fixed at a small opening at the side. The whole container includes a filling gas and does only work, when it´s affiliated to a power source.
As the Geiger counter is associated to the power source, an electric field arises. Upon the penetration of radioactive radiation through the opening, the filling gas gets ionized. In case of the electric field the ions move to the Geiger counter wall, while the electrons go to the wire. The electrons become faster and turn into an electron avalanche. This electron avalanche causes a current surge, that can be registered by the counter.