The Eastern Hemisphere of the Earth is a little-used concept because there is no obvious demarcation line separating it from the Western Hemisphere, to act the way the equator divides the Northern and Southern Hemispheres.
The most obvious, albeit arbitrary, divider is the Prime Meridian. Using it as a dividing line, the Eastern Hemisphere would cover most of Europe, Africa, Asia, and Oceania, leaving North America and South America in the Western Hemisphere.
Due to historical happenstance, the concept of Western Hemisphere is commonly used, but in a political rather than geographical sense: as a synonym for Western world, meaning the industrialized nations of North America and western Europe.