Let's start with the somewhat disturbing fact that the earth is a big nuclear reactor. At its core, the decay of naturally occurring isotopes under immense pressure produces temperatures of around 7,000 degrees Fahrenheit. This heat radiates outward through a semi-molten mantel, eventually diminishing enough to allow a solid crust to form.
From our vantage point here on the surface, the further down you go, the hotter it gets. Start drilling on a cold winter day in Siberia, and by the time the drill bit travels a few miles, it's hot enough to boil water. This heat mostly stays down there where it belongs. But every once in a while, a crack forms in the crust, and rainwater seeps down until it meets superheated rock, at which point it vaporizes and rises through a different set of cracks to the surface. The result is a geyser like Yellowstone's Old Faithful or the type of hot spring around which resorts are built.
But sometimes hot water doesn't make it all the way to the surface and gets trapped by a layer of impermeable rock, where it becomes a "geothermal reservoir." Drill into it and steam and/or hot water capable of running a turbine is released. Such reservoirs are being discovered all over the world, but they're most common in tectonically active places like Iceland and along the Pacific Rim's "Ring of Fire."
An Italian prince named Piero Ginori Conti is generally credited with building the first geothermal power plant in Italy in 1913. The first geothermal plant in the United States was built in 1962 at Geysers Field in northern California, which is still the world's largest producing geothermal field. The original technology is still in use, though the future belongs to some newer variants.
Get an overview of the main geothermal forms and processes set up to generate power from geothermal.