The stereotypical "princess hat" (at least, that's what I called the one I had when I was six) is commonly known as a hennin or henin - originally a derogatory term for a different form of headdress worn in France in the 1420s that was popularized by Herbert Norris, the famous early twentieth-century fashion historian. (Another term sometimes used is "cornet", but I'm not sure if this is what was used in the period or if it's a later term applied in a more knowingly anachronistic fashion. We frequently have a number of dress-related terms and a number of images of dressed people, and little to no way of figuring out exactly which terms correspond to which garments.) Cheunsoong Song and Lucy Sibley, authors of "The Vertical Headdress of Fifteenth Century Northern Europe (Dress, vol. 16, 1990), simply use "cone-shaped headdress" to avoid anachronism and confusion.

This headdress seems to have arisen by about 1450 in France, quite possibly as an alteration of the basic fifteenth century open hood (good pictures of a reproduction in that link) with the long, thin liripipe stiffened with wire or buckram - compare it to the view of an unveiled henin from the side in this illustration of "Prowess and Boldness addressing a group of women", from "Le chemin de Vaillance", 1473-1483; BL Royal MS 14 E II, f.103v. (On the other hand, there are other, earlier depictions that show nothing but the conical part - Scene from "Regnault de Montauban", 1440-1467; BNF Bibliothèque de l'Arsenal, Ms-5072 réserve f.385v - so that's not a definite origin.) I'm not sure anyone can give a solid date for when this fell out of fashion, but certainly by 1500 it had been supplanted by a new variation on the basic hood, the "French hood" - a fitted cap, often of colored and/or jeweled fabric worn with a black hood, such as seen in "Bathsheba", from the Hennessy Hours, Simon Benning, ca. 1535; BRB ms. II 158, f.143v. A version that was more squared off at the top was fashionable in the Low Countries during the same period: see "Tommaso di Folco Portinari and Maria Portinari", painted by Hans Memling in Bruges, ca. 1470.

The little bit of gauze often hanging from the tip of a modern costume princess hat is not accurate to any period - I would suspect someone did it in a twentieth-century film and it caught on. Women's headdresses in the later Middle Ages very frequently involved a large square of white linen that was either wrapped around the head (particularly in the lower classes, where it had a functional purpose - keeping the hair out of the way) or starched and used as a veil; in this case, as it was a fashionable garment worn by wealthy women, the linen was as thin and transparent as possible, and was often draped over the entire point of the headdress (probably with a pin in it somewhere to stay in place. In some cases, the veil was apparently suspended just from the tip, which is probably what inspired the original costume use of the little streamer.

There has been some talk of the hennin having come from Asian fashions, originally in The Secret History of the Mongol Queens (Jack Weatherford, 2011) and then taken up by the Smithsonian website via the blog MedievalPoC. Unfortunately, none of these writers were fashion historians, so they were unaware of the great disparity of time between the mid-fifteenth century - most likely thinking of this garment as a generic medieval princess hat - and the time when Marco Polo may have brought back a Mongolian boqta, around 1300.