It's a good question, as it's a very vaguely defined job title, but generally the CEO sets overall vision, and managing the board, investors, etc (i.e. external facing) - and the President focuses on execution of that vision (i.e. internal facing). President/COO are interchangeable in this case.

Think about Facebook: Zuckerberg sets direction/vision, and Sandberg executes against that. (I'm sure it's not as clear-cut in reality, but you get the idea.)

In some larger companies, there are multiple presidents with responsibility for their own divisions, who all report up to the CEO.