A sphere of influence is a metaphorical region of political influences surrounding a country. It is also known as an SOI.
When a country falls into another's "sphere of influence" that country frequently becomes subsidiary to the more powerful one, operating as a satellite state or de facto colony.
For example, during the height of its existence, the Japanese Empire had quite a large sphere of influence, with the Japanese government influencing, or directly governing events in Korea, Manchuria, Vietnam, Taiwan, and parts of China. The Japanese "sphere of influence" could thus be quite easily drawn on a map of the Pacific Ocean as a large "bubble" surrounding the islands of Japan and the Asian nations it controlled.
During the Cold War, Western Europe, Japan, and South Korea were often said to lie under the sphere of influence of the United States, while Eastern Europe, North Korea, Cuba, Vietnam, and (until the Sino-Soviet split) the People's Republic of China were said to lie under the sphere of influence of the Soviet Union.
Sometimes portions of a single country can fall into two distinct spheres of influence. In the colonial era the buffer states of Iran and Thailand, lying between the empires of Britain/Russia and Britain/France respectively, were divided between the spheres of influence of the imperial powers. Likewise, after World War II, Germany was divided into four occupation zones, which later consolidated into West Germany and East Germany, the former a member of NATO and the latter a member of the Warsaw Pact.