The Western Hemisphere contains The Americas and nearby islands. The terminology is meant to serve as an analogy with the natural geographic division of the world into a Northern and Southern Hemisphere. However, it is more a geopolitical rather than a geographical term, and refers mostly to the governments and nations that are located in the region; there is no physically-based boundary on the planet that actually separates the Western Hemisphere from the rest of the world, such as the equator which is defined by the rotation of the earth and which thus separates the Northern and Southern hemispheres. (Some suggest that the eastern and western hemispheres are divided by the Prime Meridian and the 180° line of Longitude. This would mean that parts of Europe, Africa, and Asia belong in the Western Hemisphere.) Neither is there a way to determine which of two hemispheres (wherever one draws the line between them) is the "western" one, since "western" is a relative direction. The area commonly called Western Hemisphere is western only relative to an observer in Europe or Africa, and so the term is considered eurocentric.
The term Eastern Hemisphere tends to be used much less often than Western Hemisphere.