An armet, or armet à rondelle1, is the traditional helmet seen on fancy suits of armor. These were not by any means the most common type of helmet, but they look nice and are easy to mount on a decorative set of armor, as the entire head-gear ensemble is welded into one piece.
The armet is a visored helmet that covers the entire head: chin, face, front and back of the cranium, and at least the top of the neck. It joins closely with the corselet (body armor), and provides maximum protection against slicing and stabbing. It is also a hindrance to peripheral vision, and rather constricting (not to mention hot). It was particularly used by knights that were often engaged in rather formal forms of battle, such as jousting.
It is worth noting that the great helm and the bascinet, both of which preceded the armet, also encased the entire head. In the case of the great helm it encased the head in much the same way as a giant tin can would. The armet was called the 'small helm' (see fn 1.) because it was more carefully crafted, more fitted to the form of the head, and most importantly, lighter. There are also any number of other helmets that were worn with various face and chin guards, but these guards had to be strapped on separately; while the resulting warrior might appear much like an armet-clad knight, the armet was easier to don, could have the face-plates opened without unstrapping anything, and was something of a status symbol.
There are technically two types of light helmet that encase the entire head and neck; the armet and the close helm. The difference between these two is trivial by today's standards, and it is common to refer to both types of helmets as armets. One of the defining features of the armet is the face-plates2. The close helm has the visor (sometimes a double visor) that we are most familiar with, hinged to open upwards; this what we usually see on knights in the movies. The armet proper also had the visor, but in addition had face-plates that opened on hinges, either opening up (like a welder's mask), or out in two flaps to either side3. This was quiet a useful feature, and was probably more comfortable than the close helm, but did not look very cool (by modern standards, anyway) when the face-plates were hanging open.
Armor making is serious business, and as such has a very technical vocabulary. Each plate of the armet has its own name. The skull-piece was called the calotte or timbre; this covered the scalp, over the back of the head, and down to the nape of the neck. The visor was hinged and covered the eyes and face. The chin plate was called a bevor. The neck plate was called a gorget, although often the French gorgerin was used. At the back of the neck there was a Rondel (rondelle in French), a disk set on a short peg, protruding like a giant pushpin, which was originally used as an attachment for the straps of a separate gorget and perhaps as extra protection for the seem at the back of the neck, but was later kept as an ornament, and was sometimes decorated with a scarf.
The armet appeared sometime in the early 1400s, and remained in use at least into the 1600s, although most of the documentation we have is in the form of drawings and paintings of the ruling class. As more nitty-gritty warriors moved onto lighter helms such as the burgonet, it is likely that the armet was kept in use by the wealthy just to show off how wealthy they were.
1. Some consider the rondelle to be a defining feature of the armet, making the term 'armet à rondelle' somewhat redundant. The word 'armet' probably comes from the Italian elmetto or armetto, meaning 'little helm'. It is sometimes called a heauine in French, which also literally translates as 'small helm'. The same helmets are called a Csőrsisak in Hungarian (they were popular in Hungary). The term 'armet petit' referred to an armet without the bevor, although some experts say that this sort of helmet was not really an armet; sometimes the term armet grande was used to refer to one with the bevor attached.
2. The other, of course, being the rondelle.
Groovy diagrams (text in French)
Notes on arms and armor By Bashford Dean
A cyclopaedia of costume or dictionary of dress, including notices of contemporaneous fashions on the continent by James Robinson Planché
The Encyclopedia Americana: a library of universal knowledge, Volume 14