One of the chief obstacles in john kennedy's presidential bid in 1960 was his

Religion

John F. Kennedy was Roman Catholic, and there was some apprehension that his allegiance to church doctrine and policy set by the papacy would influence how he would govern as president.  In September of 1960, Kennedy as candidate gave a major speech outlining his commitment to the separation of church and state, winning the public's confidence on that issue.  A sample of what he said in that speech:  
Because I am a Catholic, and no Catholic has ever been elected president, the real issues in this campaign have been obscured. ...So it is apparently necessary for me to state once again not what kind of church I believe in — for that should be important only to me — but what kind of America I believe in.