Basic Information
The helmet shall, for clarity, serve as a collective name for all kinds of rigid head armour. There are a variety of historical names for various styles, most of which will be cheerfully glossed over except as suits the narrative of the page.
That said, it should be noted that "helmet" is, technically, a diminutive of the word helm - which should be familiar to those aware of medieval armour. Where the difference (if any) lies between the two types is open for debate - some suggest that a helm covers the entire head, whilst a helmet only gives partial coverage, but this is not certain. The only definitive case would seem to be that of the Great Helm (possibly the opposite end of the scale to a hypothetical helmet). This was a truly enormous piece of metal work, apparently confined to jousting armour (despite the mistaken beliefs of some early scholars), not least because of the extremely limited field of vision that it affords. The great helm was essentially a metal casement that bolted to the top of the wearer's armour and inside which his head was free to move about. So free, in face, that most users tended to wear a second helmet inside to avoid banging their head on the inside of the great helm. Essentially this device sacrificed the ability to see anything that wasn't directly ahead for a greatly reduced chance of having your neck snapped by a lance hitting you on the head, or (part of) a lance forcing its way through a joint in a battle helm and killing you that way instead.
Regardless, the helmet is, then, a piece of rigid head protection - usually for use in combat (the non-combatant version tends to be called a hard hat) - regardless of exact design or era. Metal is a normal material of construction (from Sumerian copper to modern manganese steel), but other materials have also been used historically including boiled leather and wood. Modern helmets tend to use some kind of plastic - ballistic or otherwise. Non-rigid head protection is generally termed a cap or coif and these have, historically, been worn under helmets on numerous occasions. Modern "semi-rigid" pieces are an unecessary complication given that they appear not to be used in combat.
The helmet may be fitted to the user with an interior web of straps, or more simply attached by strapping onto a layer of padding (perhaps including a coif). Direct, rigid contact with the skull is normally a bad idea as it makes it too easy for impacts to propagate directly into the brain - and is also a very uncomfortable way of wearing a heavy piece of metal.
The outside of the helmet may be adorned by mounting a plume on top, or camouflaged by the fitting of a cloth cover - or pretty much anything in between. Dramatic structures like horns or wings were rare on actual fighting helmets as they constituted an encumbering nuisance, and when present tended to be lightly attached, cosmetic pieces. Any helmet with integral metal horns is almost certainly a ceremonial artifact. Extravagant crests were also a feature of jousting helms, partly as a display element and partly as an additional scoring opportunity.
Other design features include the possibility of a brim (typically found on infantry helmets designed to protect from plunging fire) and face protection. Face protection in particular could range from none at all to a closed face panel with minimal eyeholes. In between lay a wide range of flexible and non-flexible protections, including ornate masks, mail veils and a range of bars and cross bars. In a lot of cases all or part of the face protection folded out of the way - a visor if if covered the top or all of the face1 or a bevor2 (although this term can also apply something more like a gorget). A helmet might also be fitted with pteruges to protect the neck or an aventail (also called a camail) - which was a sort of skirt of mail hanging down from the rim. Failing that the wearer might need to rely on a gorget or pixanne (a mail garment covering the neck and shoulders), or even some kind of stock or other collar.