Why are there lots of earthquakes around the rim of the Pacific Ocean?

To explain the incidence of earthquakes around the Pacific Rim, we begin one of the central concepts of geology: that of plate tectonics. The Earth’s crust of solid rock is a relatively thin layer (5-50 km thick), while below there is the thick, hot layer known as the mantle. The mantle, although very hot, is considered solid, owing to the great pressure on it. Although it is considered solid, it deforms and flows on long time scales. This allows the Earth’s crust to slowly move. Huge sections of the crust, called tectonic plates, stay together but move relative to other plates. Many plates are continent-sized, but some are smaller, and most move about 1-2 cm per year.
Interesting things happen at the boundaries between crustal plates. They move relative to one another, so in some places they are moving apart, in some they are converging, or moving together, and in some they are sliding past one another, going in opposite directions. Where plates are moving apart, new crust forms from the upwelling of mantle rock to fill in the gap. As hot mantle rock approaches Earth’s surface and experiences less pressure, it melts and the resulting lava wells up and forms new crust. This is occurring in Africa’s Great Rift Valley and in the mid-Atlantic ridge, of which Iceland is a part.
The Pacific Ocean sits mostly on top of a single plate, the Pacific Plate. Most of the boundaries of the Pacific Plate are convergent, meaning plates are moving toward one another. Essentially, all of the surrounding continents are encroaching on the Pacific Plate. Something has to happen to the crust at a convergent boundary! When oceanic crust meets continental crust, it subducts beneath it. That means the crust of the ocean floor bends downward, and as the plates continue to push together, it moves down and down into the mantle and becomes part of the mantle rock. Some areas around the Pacific feature plates moving sideways with respect to one another. The San Andreas fault system in California and Mexico is this kind of boundary.
At all types of plate boundaries, we have rock pushing past other rock that is moving in the opposite direction. Most of the time, friction keeps the solid rocks from moving easily past one another. The rocks remain connected but bend, or deform, as rock to which they are attached continues moving. Strain builds up as the rocks are deformed more and more. When the energy associated with the strain is greater than would be needed to break the rocks outright, the rocks break. Significant motion occurs and a lot of energy is released as rocks “relax” to their undeformed state. This is an earthquake.
In short, the reason there are frequent earthquakes all around the Pacific Rim is that the boundaries of the Pacific Plate underlie this region, and plate boundaries are where earthquakes typically occur, because of the buildup of strain as rocks move in different directions.