Like smoke detectors, Geiger counters depend on a flow of the electrical current to detect radiation.
There are two types of smoke detector commonly available in many countries. One type uses americium-241, a source of alpha radiation, to detect the presence of smoke or heat sources. Here's a diagram showing how it looks inside.
The other type of detector does not contain radioactive material; it uses a photoelectric sensor to detect the change in light level caused by smoke.
As for the Geiger counters, they are used to detect radioactive emissions, most commonly beta particles and gamma rays.
The counter consists of a tube filled with an inert gas that becomes conductive of electricity when it is impacted by a high-energy particle. When a Geiger counter is exposed to ionizing radiation, the particles penetrate the tube and collide with the gas, releasing more electrons.
See how Geiger counter works from this link: http://atomic.lindahall.org/what-is-a-geiger-counter.html.