A sphere of influence is a metaphorical region of political influences surrounding a country.
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.
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 Great Britain/Russia and Great Britain/France respectively, were divided between the spheres of influence of the imperial powers.