Chateauesque is exactly what it sounds like: it’s a style based on sixteenth-century French chateaus in the Loire Valley of France. The elaborate revival style became popular between the 1880s and 1900s and was favored by the affluent.
The Chateauesque architecture style is rather easy to identify. Its main characteristics are:
Round tower with a conical roof
Steeply pitched hipped or gable roof
Masonry (stone or brick) construction
German Renaissance Revival and French Renaissance Revival are closely related (and often used interchangeably) with the Chateauesque. The main difference is that while the Chateauesque buildings can be asymmetrical, French and German Renaissance Revival structures are symmetrical and well proportioned.
The style was made popular in the United States by Richard Morris Hunt, the first American architect to study in the École des Beaux-Arts in Paris.
The Dakota is a mix of Chateauesque, aka French Renaissance and German Renaissance.