A wide variety of hats and caps were worn throughout Elizabeth's reign; each could indicate something about your role in society and, of course, how fashionable you were. Like many aspects of Sixteenth Century life, one might be obliged to wear a particular type by mandate.
In 1571 a statute called An Act for the Continuance of the Making of Caps determined that “… all [males] above the age of six years except some of certain state and condition, shall wear upon the Sabbath and Holydays, one cap of wool knit, thicked [felted or fulled] and dressed in England, upon the forefeiture of 3s 4d …”
Though the selection of headwear is vast, there are at least two looks (that don't fall into any of the categories below) often worn by reenactors that don't seem to have been done in the period. The first is the wearing of a white linen coif or Biggins alone or underneath other hats. This may have been done by children and old men, but it appears to have been very uncommon among men in general. The other is the wearing of what are often called "Riding Hats" today; the modern versions of these are usually fabric-covered and similar to an arched top hat with a curved brim in shape. Though they are very popular among men at Renaissance Faires, in the period they seem to have been all but exclusive to women. One example can be found atop the head of an Englishwoman here in a drawing by Lucas de Heere from 1570.
Worn from the early part of the period on, Flatcaps appear to have decreased in popularity after the third quarter of the century. These were almost always knitted from wool and fulled, rather than made out of sewn fabric like velvet or wool. The solid fabric type, called Bonnets, don't seem to have been anywhere near as common as the knit type, but they did exist. They might be constructed of equally sized-pieces of fabric or with a larger crown pleated into the brim. The brims of Flatcaps could be quite decorative, ranging from a two-piece "split" brim like in the top right image below, or even slashed all the way around like in the example from the Museum of London below it. I have heard that apprentices and/or workingmen were required to wear these, but I can't find the source at the moment.
Hats made of wool or fur felt blocked or set set into shape became increasingly fashionable as the 16th century wore on. They first appeared in the 1560's, but really seem to have taken off during the following decade in England. Some hats had a pile, or a raised texture of fabric or yarn, but smoother finishes dominated. In England, they tended to be very tall (seemingly at least around six inches or more in height), unlike in central and Northern Europe where low crowned hats share the field. The top was usually domed, but examples of hats with tapered flat tops do show up towards the 1590's and beyond.
One of several things might be done with the brim; it could be left alone, turned up on both sides of the head, turned up in the front, or turned up on only one side of the head (the right or the left). All of these are documentable. The practice of turning it up on only one side of the head may have been done from time to time, but examples are very very rare in primary sources as compared to the other methods. Despite this, many reenactors jump to this style (usually with a low crowned hat) and thus they are overemphasized and misrepresented to the public.
Shades of black, brown, gray, tan, and white seem to be the most common colors. Some might have been lined like the example from the "Fete at Bermondsey" below.
Monmouth Caps likely originated in Wales at some point in the late Middle Ages. They were fully knit from wool, unlike modern beanies and caps that are usually partially knit and then sewn together at the top. The surviving Sixteenth Century example from (fittingly!) The Monmouth Museum features a knit button on top and a loop at the back. These could be means of finishing off the yarn or perhaps one or both are decorative. These were considered to be very practical, and it was even recommended by the (real) John Smith of "Pocahontas" fame that new colonists heading to Virginia bring one along. Ir is also possible that Monmouth Caps at one point or another became the defacto issue or choice for soldiers. A line from the late Sixteenth or early Seventeenth Century Ballad of the Capsreads "...The Souldiers that the Monmovth wear..." Many records of garments issued to soldiers in the period mention caps, but none that I have seen to date specify whether they are a Flatcap or Monmouth Cap.
Thrum Caps appear to have been very shaggy caps similar in shape to Monmouth Caps. Knit from wool, the texture likely came from additional threads or bits of yarn pushed back through the weave of the knitting to create a very thick wall of warmth for service at sea. It is possible that these were felted or fulled for additional warmth and protection against the wet as well.