In political science, the Head of State is the person who represents a nation, and who is considered to be the most important official in the government. The Head of State is often a ceremonial role, which differentiates them from the Head of Government, who often has more day-to-day control.
What exactly the Head of State is, like many questions in political science, is often hidden behind tradition and vague definitions. The concept comes out of the tradition of a monarch, and even in a Democracy or Republic, the Head of State is meant to reflect the idea of a leader, even a parental figure.
Any decent reference book, or even common sense, would tell us that the Head of State in the United States of America is the President of the United States of America. The President of the United States is the most prestigious office, the most competed for, and the person who in many ways has life or death power over matters of foreign policy. So, by wide consensus, the President of the United States is considered to be the Head of State of the country, the ceremonial leader. His theme is Hail to the Chief, after all.
And yet, reading the Constitution of the United States, the President's role of "Head of State" is not clearly stated. The term and concept may not have yet existed. The President is vested with many positions of authority, including being the Commander-in-Chief of the military, appointing other executive officers, pardoning people, and giving a speech. While some of these are very important powers, nowhere does it say that the President is the overall "Head of State". In fact, the powers and duties of Congress are listed in much more detail, and also are listed in the First Article of the Constitution, as opposed to the Second Article. In other words, the constitutional duties and powers of the Presidency are much more limited than they might generally be considered to be, and the constitution doesn't grant the Presidency any sort of primacy amongst the branches. Even the Presidency's traditional dominance in foreign policy is based on the President's ability to make treaties, and to "receive Ambassadors".
In other words, the President is "Head of State" only by the weight of tradition and public perception, but not through any actual statement of the Constitution. The President is legally only the possessor of certain modular power, and any extrapolation of that into the President being a symbol of the nation's sovereignty has no legal basis. This has in fact been established by the Supreme Court in Youngstown Sheet & Tube Co. v. Sawyer, where it was decided that the President only has powers enumerated in the constitution, or granted to him by congress.
This being the case, the President still has a great deal of prestige, both domestic and foreign, and also, through the years, has become more powerful that his strictly enumerated duties would suggest. Therefore, while it is not strictly true that the President is the "Head of State", it still makes sense to refer to him as such.
Sexist language used throughout because at this point, all American Presidents have been male.