Bycocket Shaped Hat

This hat is shaped like a bycocket or robin hood hat, made of patterned fabric, lined and decorated with a button and feathers.

The hat is made of soft fabric which makes it sit very differently on the head, it flattens out spreading outside the circumference of your head. Without that stiffness or a band to grip the head, it also feels more insecure than it should.

This is one of those awkward hats which I'm not sure where I will use it, it's not general-purpose to go about walking in, does not fit in a fantasy or historical use, not silly enough to be a novelty hat.

Currently thinking it may get used for some sort of drippy nobleman character

This was a Christmas present, much appreciated even if unsure of use yet.

Reversible hat

The fabric hat is reversible so you can a brown or red hat 

I would hazard a guess that the hat is a medieval style, the kind that would normally be made from wool.

The hat is basically a square bag, that follows the shape of the head when worn and leaving 4 corners at the crown where there is an air pocket above your head.

Many shapes of historical hats do not have names, you could call it a beanie of shorts

I would say this hat suitable for any period from at least the 13th to 18th century. 

Mob Hat

 A Mob hat or cap was worn by Georgian women of any status. 

Its also called a Mop cap which is probably due to a subtle change its use from a frilly bonnet to a hat used by victorian maids who mopped the floor.

This style of hat is mainly associated today with Victorian maids and would normally be white.

Basically, a round cloth gathered with a ribbon or string to give a ruffle brim and covering.

Designed to keep the hair under control, modern plastic and net variants are used in clean environments such as food factories for the same reason.

These caps or a version of them were worn by men at least in the early 18th century but were predominantly women headwear.