The Ring of Fire is a ring of volcanoes around the Pacific Ocean that result from subduction of oceanic plates beneath lighter continental plates.
Subduction of oceanic lithosphere.
Most of the Earth's volcanoes are located around the Pacific Ring of Fire because that the location of most of the Earth's subduction zones. A subduction zone is a place where one plate of oceanic lithosphere (= the crust + uppermost mantle) is shoved under another plate. The downgoing plate is always the oceanic one. All while it was oceanic plate it collected water-saturated sediments and its uppermost few hundred meters got water saturated also. As it is shoved into the hotter mantle the plate heats up and all this water and other volatiles boil off and migrate upwards through the overlying plate. The addition of volatiles such as water to the hot overlying mantle causes partial melting and the production of magma. This magma rises up through the over-lying plate to erupt at the surface. If the overlying plate is a continent, you get a chain of volcanoes such as the Andes or Cascades. If the overlying plate is ocean you get a chain of volcanic islands such as the Marianas or Aleutians. This is also where the Earth's deep ocean trenches are and where the Earth's deep earthquakes are. The trenches form because the downgoing plate is bent downward as it subducts. The earthquakes form as the two plates scrape against each other (earthquakes down to about 150 km) and then as the downgoing plate bends (earthquakes down to about 700 km). The earthquakes do a very good job of tracing the position of the downgoing plate. These zones of earthquakes are called Wadati-Benioff zones, after the two seismologists who first recognized them.
Wikipedia has an excellent overview of the basics here.