In stark, revealing contrast to the glamorous, effortless Kennedy of popular myth, John F. Kennedy on the stump was a tough, savvy campaigner. Well aware that much of the country distrusted almost everything about him — his Massachusetts-liberal politics, his Boston accent, his Roman Catholicism — like any good politician he set about winning over skeptics by employing the very gifts that generated such suspicion in those who knew little about him. He charmed. He cajoled. At times (in back-room negotiations with other pols) he browbeat and he bullied.

[Buy the LIFE book, The Day Kennedy Died.]

And on November 8, 1960, John Kennedy was elected president of the United States, defeating Richard Nixon in one of the closest national elections of the 20th century. At 43, Kennedy was (and remains) the youngest person elected to the office, and it was largely this quality in the man and his family — an engaging, youthful dynamism — that so captured the imagination of millions across the country and, ultimately, across the world.

As Kennedy and his team ran a heady, propulsive campaign unlike any America had seen, LIFE’s best photographers (Paul Schutzer, Alfred Eisenstaedt, George Silk and others) were there, chronicling the grind of never-ending public appearances and the quieter moments JFK spent with advisers, with Jackie and — rarest of all — alone, with his own thoughts.

Here, LIFE.com presents photos — none of which ran in LIFE magazine — chronicling an enigmatic, intensely ambitious man making history.