It is hard to know which roles you are thinking about since you have only told us two of the ones that you know about.
In general, the president's powers have increased a great deal over the years. For example, the president is now seen as the "Legislative Leader" or "Chief Legislator" even though the founders intended for Congress to take the lead in this area. The president is now seen as the "Manager of the Economy" and is held responsible for how well the economy goes. These are roles that have been added to the president's responsibilities since the time of the Founding Fathers.
It is unclear from your question whether you are referring to the United States or to all countries in general. Not all presidents around the world have the same powers, of course. In addition, the large list of United States presidents have exhibited different degrees of power in different areas.
That being said, in addition to the expert ideas presented in the former answer, it's also important to mention a president's role as the Commander-in-chief, a simple title for the supreme ruler of a country's military forces. In the United States, this power is given to the president by the Constitution. The head of a military force as vast as the United States Military (with its many branches) is granted a lot of power, indeed.
There are many examples of presidents acting as Commander-in-chief, but one of the more poignant and recent examples occurred when President Barack Obama (of the United States) gave the order for the Navy Seals to enter Osama Bin Laden's compound in Abbottabad, Pakistan. Bin Laden, of course, was killed as a result.