The Ring of Fire
The Ring of Fire (also known as the Pacific Ring of Fire, the Rim of Fire, the Girdle of Fire or the Circum-Pacific belt), is a tectonic belt of volcanoes and earthquakes. The word tectonic refers to physical movements and chemical changes over time in the Earth's crust, its uppermost layer. It is about 40,000 km long and up to about 500 km wide and surrounds most of the Pacific Ocean; see figure.
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The Ring of Fire contains between 750 and 915 active or dormant volcanoes, around two-thirds of the world total, so you can understand where its name came from. About 90% of the world's earthquakes, including most of its largest, occur within the belt.
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The Ring of Fire is not a single geological structure. To understand it,
we need to learn some new words in the field of Geology or Earth Science.

Lithosphere
The rigid outermost part of Earth (or any planet) is called the Lithosphere. In Earth, this consists of the outer crust and the upper mantle; see figure. Oceanic lithosphere is crust that exists in the ocean basins and is denser than continental lithosphere.
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Plate Tectonics
Earth's lithosphere is fractured into seven or eight major plates (depending on how they are defined) and many minor plates or "platelets"; see figure.
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Continental drift was a theory that was proposed about a 100 years ago, that Earth's continents move or drift relative to each other over geologic (very large) time scales. This theory was rejected because people could not figure out where the energy for such large movement could come from.
Later on, after advances in physics, scientists realised radioactivity of naturally occurring elements like uranium (U238) and thorium (Th232) and potassium (K40) could generate enough heat. Along with the primordial (meaning original) heat when the Earth itself was formed, this heat energy, amounting to about 45 Terawatts, flows constantly from Earth's interior to the surface. This massive heat is sufficient to power the movement of the continents, and is called mantle convection.
Today, the theory of plate tectonics, has proved that the Earth's lithosphere is made of several tectonic plates which are actually moving (and so the continents that "sit" on these plates are also moving), and have been slowly moving since 3-4 billion years ago. Plate tectonics came to be accepted after it was validated in the mid to late 1960s. The processes that result in these plates and shape Earth's crust are called tectonics. The relative movement of the plates typically ranges from zero to 10 cm annually.

Plate boundaries
Where the plates meet, their relative motion determines the type of plate boundary (or fault). There are three types: convergent, divergent, or transform; see Box for details.
BOX: Types of Tectonic Plate Interactions
Convergent boundaries are areas where plates move toward each other and collide. These are also known as compressional or destructive boundaries.
Divergent boundaries are areas where plates move away from each other, forming either mid-ocean ridges or rift valleys.
Transform boundaries occur when two plates grind past each other with only limited convergent or divergent activity.
END OF BOX
At convergent boundaries, two continental plates can collide and push upwards to form large mountain ranges such as the Himalayas. Another type of convergent boundary is a Subduction zone where an oceanic plate meets a continental plate and is pushed underneath it.
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In general, if two plates meet, the heavier one will be pushed below. Since ocean lithosphere is more dense, it is the one that is pushed below. So the oceanic lithosphere becomes a part of the Earth's mantle as it is pushed down; see figure. Subduction zones are marked by oceanic trenches. The descending end of the oceanic plate melts and creates pressure in the mantle, causing volcanoes to form. The figure shows the subduction zones over the entire Earth.
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This is what mostly happens in the Ring of Fire. It was created by the subduction of different tectonic plates at convergent boundaries around the Pacific Ocean. These include:
the Antarctic Nazca plate and Cocos plates subducting beneath the South American plate;
the Pacific and Juan de Fuca plates subducting beneath the North American plate;
the Philippine plate beneath the Eurasian plate; 
and a complex boundary between the Pacific and Australian plate.
The interactions at these plate boundaries have formed oceanic trenches (depressions in the ocean floor), volcanic arcs (a belt of volcanoes on the surface above the subducting plate, parallel to the trench and further away), back-arc basins (under water) and volcanic belts (regions of volcanic activity where magma is created by partial melting of solid material in the Earth's crust and upper mantle due high temperatures produced in the collision). In short, it has created the Ring of Fire.
The Ring of Fire has existed for more than 35 million years but subduction has existed for much longer in some parts of the Ring; many older extinct volcanoes are located within the Ring. More than 350 of the Ring of Fire's volcanoes have been active in historical times while the four largest volcanic eruptions on Earth in the Holocene epoch (current geological epoch) all occurred at volcanoes in the Ring of Fire. These occurred at Fisher Caldera (Alaska, 8700 BC), Kurile Lake (Kamchatka, 6450 BC), Kikai Caldera (Japan, 5480 BC), and Mount Mazama (Oregon, USA 5677 BC). Notice that we are mentioning the countries where these are currently located, which may not have existed at the time of eruption!
Most of Earth's active volcanoes with summits above sea level are located in the Ring of Fire. So both volcanoes and earthquakes are common in this area.
So the next time you hear of the Agung, Sinabung or Dukono volcanoes in Indonesia, or the Mount Ruapehu volcano erupting in New Zealand, or earthquakes in Japan or California, you will know that the Ring of Fire is living up to its name.
Sources: Wikipedia, also for the images