What is the name of the region of charged particles trapped by Earth's magnetic field?

The region of charged particles trapped by Earth’s magnetic field is called the Van Allen radiation belts.

Earth has two principal belts: an inner belt dominated by energetic protons and an outer belt containing mainly high-energy electrons. The particles are held by Earth’s magnetosphere, spiraling along magnetic field lines and bouncing between regions near the North and South Poles. Their paths also drift around the planet.

The belts are not the same thing as the magnetosphere, ionosphere, or plasmasphere. The magnetosphere is the much larger magnetic environment that confines and shapes the particle population. The ionosphere is a region of the upper atmosphere containing ionized gas, while the plasmasphere is a cooler, denser plasma region inside the magnetosphere.

James Van Allen’s instruments aboard Explorer 1 helped reveal the belts in 1958, and subsequent spacecraft mapped them in greater detail. Solar storms can inject and accelerate particles, temporarily changing the belts and sometimes producing an additional short-lived radiation zone. The belts protect the atmosphere from some incoming energetic particles but create a serious radiation hazard for spacecraft and astronauts.