- “A” is used when we're talking about only one thing of various.
i.e.: A car, A house, A senator, A piece of pie
- “The” is used to refer to something unique.
i.e.: The moon, The king, The water, The Sahara desert
Since there is only one president in the U.S.A., the right one is “The President”. It would be a different case if we were talking about a senator or a candidate for presidency, because there are more than just one of them.