Short answers, pulled from the story.
A sphere of influence is a geographic region or set of nations over which a more powerful state or organization holds cultural, economic, military, or political influence. Formal treaties are not required for a sphere to exist; influence can operate through hard or soft power. Historically, strictly enforced spheres have been associated with higher levels of global conflict.
President James Monroe, in office from 1817-25, formalized the Monroe Doctrine, asserting that the New World was to be part of the American sphere of influence and removed from European encroachment. Alexander Hamilton had earlier outlined the underlying ambition in the Federalist Papers. As recently as 2018, U.S. Secretary of State Rex Tillerson invoked the doctrine in discussions about American versus Chinese trade influence in the Western Hemisphere.
A secret protocol attached to the Molotov-Ribbentrop Pact of 1939, revealed only after Germany's defeat in 1945, divided northern and eastern Europe between Nazi and Soviet spheres. Finland, Estonia, and Latvia were assigned to the Soviet sphere; Poland was to be partitioned along the Narev, Vistula, and San Rivers. The USSR denied the protocols existed until the Soviet Union dissolved, at which point the Russian government fully acknowledged their authenticity.
During the mid-19th and early 20th centuries, Britain, France, Germany, Russia, and Japan each secured spheres of influence across large areas of China by forcing the Qing government to sign unequal treaties. Germany took Shandong, Russia gained territory north of the Great Wall, France received Yunnan and parts of Guangxi and Guangdong, Japan took Fujian, and Britain claimed the entire Yangtze River valley. The system ended after World War II.
In November 2014, German Chancellor Angela Merkel said that "old thinking about spheres of influence, which runs roughshod over international law" put the "entire European peace order into question." In January 2017, British Prime Minister Theresa May warned against accepting President Putin's claim that Eastern Europe now lay within his sphere of influence, calling it a jeopardizing of the freedoms brought to the region by President Reagan and Mrs. Thatcher.
In corporate terms, a sphere of influence describes how a business shapes the decisions of other organizations through its market size and reach. Microsoft's dominance in operating systems means software sellers must consider compatibility with its products. Wall Street spent a record $2 billion trying to influence the 2016 United States elections, illustrating how corporate spheres extend into regulation and politics.