At a tumultuous meeting of anti-Vietnam War militants at the Chicago Coliseum in 1969, Bill Ayers helped found the radical Weathermen, launching a campaign of bombings that would target the Pentagon and United States Capitol. Twenty-six years later, at a lunchtime meeting about school reform in a Chicago skyscraper, Obama met Mr. Ayers, by then an education professor. Their paths have crossed sporadically since then, at a coffee Ayers hosted for Obama’s first run for office, on the schools project and a charitable board, and in casual encounters as Hyde Park neighbors. Their relationship has become a touchstone for Obama’s...