The President of France is the elected head of state. France has gone through five separate republican governments. The Second through the Fifth (the current republic) had a president as the head of state. Thus, arguably, the French presidency is the oldest in Europe, if not the most consistent in creation.
Under the Constitution of the Fifth Republic, the French President wields a great deal of actual power, unlike his counter-parts throughout Europe. However, under the Third and Fourth Republics, the Presidency was a ceremonial office. The Second Republic was short-lived, and its one president, Louis-Napoleon Bonaparte, abolished the republic and proclaimed himself emperor of the Second Empire as Napoleon III.
Thus, the relevance of the President of France in the works of Harry Turtledove may vary depending on the era in which a given work is set. In The War That Came Early, for example, the office of the presidency isn't even referenced, with the head of government Edouard Daladier being the most important French political figure in the series.