Richard M. Nixon, the Republican candidate, seemed to be the likely winner of the 1960 presidential election.  He was a known entity, had served as vice president under Eisenhower, and was a Protestant.  John F. Kennedy, on the other hand, was not as well known, had no previous federal government experience and was an Irish Catholic.  (Never had there been a Catholic president!)  During the debate, Kennedy was comparatively relaxed, confident, answered the questions thoroughly, was engaging and was visually appealing to viewers.  Nixon, on the other hand, had a "5:00 shadow" (unshaven), looked tired, appeared less confident and was visually less appealing.  Kennedy was the clear winner of these debates and, as some historians believe, consequently won the election.

I wonder if you might be talking about the first televized presidential debate, not election.

If so, that was the debate between John F. Kennedy and Richard Nixon in the 1960 election.

In this debate, the power of television and the power of the way candidates looked was really demonstrated.  Nixon looked nervous and had not shaved well enough so he didn't look so good.  (He was uglier than JFK too.)

The effect was so strong that people who watched the debate said JFK won while people who only heard it on the radio thought Nixon had won.

This is believed to have helped JFK win -- especially since it was a very close election.

It happened in the 1960 Presidential election between John F. Kennedy and Richard M. Nixon.  Some historians say it changed the course of the election and history, though we'll never know for sure if that's true, or what the exact effect with voters was.

Kennedy was young, attractive and vibrant, the voice of a new generation, a new direction.  Nixon represented the experience of the past, of anti-communism and the 1950s. 

The debate was a good one, with Kennedy giving more eloquent and smooth answers, while Nixon had more experience.  The main effect we know of was that voters who listened on the radio tended to vote more for Nixon.  People who saw the debate on television tended to vote for Kennedy.  That is the visual of Kennedy contrasted with Nixon's age and appearance made JFK seem a better choice.

Did it change the outcome of the election?  Hard to say, but the difference in the popular vote between the two was a mere 118,000, and 46 million Americans watched the TV debate.