As with any society, a leader is always present. This is true even is gangs and clans. There is always someone who the rest look to either for guidance.
This is usually a person with wisdom or a knowledge base that nearly everyone can benefit from.
What I understand a President to be is mostly the leader of the military. The President is elected by civilians and is replaced every 4 or 8 years. So as not to become too strong or become a dictator.
A dictator assumes and demands complete control over every aspect of his subjects. An elected government, governs the people by the consent of the people. In other words if the current leadership is not working it are no longer considered valid and then the people place new people in office.
That is the short of it.
Here in the U.S he is the Commander in Chief. However he is not in complete control without checks and balances. In other words he can't just declare war without the consent of Congress. Which is also supposed to be by the consent of the people.
The position of the Presidency is much more than a individual. It is an institution. A President can make suggestions that he/she would like to see take effect. But he can not enforce his bills without them being adopted and voted on within the House of Representatives and Congress. Both must agree and pass a bill before it becomes law.
i.e. The health care bill that will destroy this country if left in place.
In this case, we the people of the U.S. seem to be voting out this election, many those people who did not listen to the voice of the people who they governed. Now they are losing their office. They passed this law without the consent of the population. They feel they know what is best for us and treated us like infants, imposing their opinions and beliefs on the public.
In reality they voted to pass a law that will ensure them gaining the votes of those who have become or will be forced to become dependents upon them for their needs. So far there are more of us who are not dependent upon the government and would rather keep it that way.
The Presidency is no different. This office is also the scape goat if anything goes wrong. Whether the person is to blame or not, it falls on their shoulders.