There are several different separate pieces of armor that can each cover a different part of a dwarf's body and provide protection. In Dwarf Fortress, armor (as so many things) is not a simple formula where a single, bigger number replaces a smaller one. Different armor pieces work slightly differently to provide more or less protection in different way. The exact piece, quality, and material all factor in, at least - combat is not fully understood. However, overall, rest assured that, factoring in the known variables of quality and material, your intuition will generally be correct - pieces of chain armor are better than the equivalent pieces of leather armor, and plate pieces better than chain.
Also, any soldier or hunter will have one of 4 levels of armor designated: clothing, leather armor, chain (armor), or plate (armor). There is usually not a 1:1 relationship between every piece of armor and an armor level - it can be very much a "mix and match" situation.
For a general discussion of how armor works and some of the game mechanics behind it, including charts on quality and material, see Armor.
 Clothing vs Armor
Practically speaking, there is no difference. Any piece of clothing or armor provides some level of protection against damage, including damage from heat and cold weather. Some materials have a better protection value than others - metal is better than leather (go figure). However, any dwarf, when activated into military service, can be set to one of 4 different armor levels - clothing, leather, chain or plate - and will seek out a complete "set" of that level of protection. In this sense, some pieces - like cloth vs. metal caps, or boots - fall into different categories, clothing vs armor.
Civilians will wear leather caps and boots, but no other piece of armor, with one exception. A civilian with the hunting labor enabled will seek to wear the level of armor designated - but they will also attempt to actively hunt any huntable wildlife on the map. (Only if there is none, they will seek another job, until there is a valid target.)
 Levels and Sets of Armor
In seen in either the Hunting). This is designated by viewing a dwarf and selecting the soldier menu - Clothes, Leather, Chain or Plate. (In the military weapons menu, these are abreviated as " - " (none), Lth, Chn or Plt, and also whether or not to use a Shd or Bkl - shield or buckler). This defines the maximum class of protection they will seek to wear for each body part (but they may not completely succeed, depending on available items).menu or an individual dwarf's references for " oldiering & hunting", armor can be one of 4 overall levels of armor, as defined by the game and using the terms Clothes, Leather (armor), Chain (armor) and Plate (armor), in reference to the maximum protection a dwarf will seek to wear when activated into the military (or when
 Armor Levels
The leather armor set:
- Head: leather or metal cap or leather, bone, or shell helm
- Upper Body: leather Armor
- Lower Body: leather, bone or shell Leggings
- Feet: leather High or Low Boot
- Hands: (none)
The chain armor set:
- Head: metal Helm
- Upper Body: Chain mail
- Lower Body: metal leggings
- Feet: metal High or Low Boot
- Hands: metal Gauntlets
The plate armor set:
- Head: metal Helm
- Upper Body: Plate mail
- Lower Body: metal or boneVerify Greaves
- Feet: metal High or Low Boot
- Hands: metal Gauntlets
If a dwarf cannot find a piece of the appropriate protection level, a lesser piece will be accepted. Quality or material are not taken into consideration, so for plate level armor, a no-quality copper plate mail will always be selected despite a far preferable masterful ☼steel chain mail☼ being available.
Dwarves will occasionally keep a lesser piece on when told to wear better armor and put the better armor over that - see layering.
 A quick note on weight
A quick glance at the encumbrance values shows that any dwarf can carry 2000Γ, a Strong 3000Γ, Very Strong 4000Γ, and so on. For each increment of 1000 over the limit, a dwarf's speed is very slightly hurt (maybe 5%, very approximately). This affects movement, actions during fighting, and firing a crossbow, but a high Agility counteracts this directly, adding to speed in all cases.
A full set of chain level armor plus weapon weighs about 2500* (requiring Strong). Plate level and weapon weigh about 3500* (Very Strong). A metal crossbow and full stack of metal bolts can weigh close to 1000* by themselves (compared to over 600* for an axe and ~300* for most others), adding another level of Strong to the requirements to avoid being slowed.
- (* This is iron or steel - bronze weighs about 5% more, copper 14% more. Bone weighs about 1/7 the weight of steel, leather ~1/4.)
So, more Strength and/or Agility is desirable, at least until your dwarves have mid-levels of the armor user skill. However, even if your dwarf is 3000 over-burdened, the added protection is almost certainly worth it unless you are concerned about chasing down opponents who are fleeing from your dwarves' righteous vengeance.
Bone, shell, leather, and wood all have a modifier of 50%, making them half as protective as iron before quality levels are applied. Masterwork non-metal armor is as protective as basic quality iron armor, which is respectable (if hardly optimal).
Shells can be used to make helms, gauntlets, and leggings. Bone can be used to make helms, gauntlets, leggings, and greaves. Leather can be used to make a full set of (obviously) leather level armor, as well as bucklers and shields. Wood can only be used to make bucklers and shields (elves sometimes sell other pieces of wooden armor, but they are too large for dwarves).
Metal armor is heavier than other armor, but it is significantly more protective; making it for your military dwarves is a good idea. A full suit of chain- or plate-level armor, including a buckler or shield, can be made from metal, but not all metals can be used to make armor (barring a strange mood). In order of increasing effectiveness (and decreasing weight), the armor-grade metals are copper, bronze (and bismuth bronze), iron, steel, and adamantine. Quality can also make up to a 100% difference in protection, but material is constant - see this table for more info.)
Bone and shell armor is made using the bone carving labor at a craftsdwarf's workshop. Leather armor is made using the leatherworking labor at a Leather works. Wooden bucklers and shields are made using the carpentry labor at a Carpenter's workshop. Metal armor is made using the armorsmithing labor at a metalsmith's forge or magma forge.
 Armor pieces
 Head protection
A Cap can be made of cloth, leather or metal, and can be considered a piece of clothing or armor, depending on that material. Caps made of metal are considered armor instead and are worn as part of the leather armor level.
Civilian dwarves will never equip caps except those they arrive with, probably due to a bug.
A Helm is a piece of armor covering the head. It can be made from leather, metal, bone or shell. Helms made from leather, bone, or shell are part of the leather armor level, while those made from metal are part of the chain and plate level.
Like with other armor, a soldier will pick a helm of a lower armor level if none of the designated level is available.
 Torso protection
 Leather armor
Leather armor is the least protective but simplest of the armors. It is also light, and so does not slow down a dwarf who lacks the strength or armor user skill for heavier armor levels. It is made, obviously, from leather. It can be worn as part of the leather armor level or under chain mail (but not plate mail).
 Chain mail
 Plate mail
Plate mail is made from three metal bars. It is twice as heavy as chain mail, but provides even better protection.
 Hand Protection
Gauntlets are always made in pairs, one left and one right, but the quality of each gauntlet in a pair is not guaranteed to be matched to the other.
 Leg protection
Leather, shell, and metal leggings are made from a single material (a piece of leather, a shell, or a metal bar, respectively). Bone leggings require 3 stacks of bones, regardless of the stack size.
Note that the specific type of leather, bone or shell (the creature or fish that was the source of those materials) has no effect on the quality of protection provided, only on the final value of the leggings manufactured.
Greaves are a heavier form of protection for the legs, and are plate armor level protection. The term "greaves" is like "pants" or "leggings" - they come as a set, you only need one "greaves" per dwarf.
They can be made from metal and bone but not from shell. "Greaves" are a single item, like "leggings" or "trousers" - for one set of bone greaves you need 3 stacks of bones, no matter how many single bones are in the stacks; for one set of metal greaves you need 2 metal bars. Even though bone greaves are part of the plate armor level, they are in between copper and bronze leggings in terms of protection.
 Foot protection
 Boot, low or high
A boot is a type of shoe that only military dwarves will wear. They come in low boot or high boot varieties - high boots weigh more than low boots, but offer better protection. Lightweight boots can be made of leather at the leather works as part of the leather armor set. Boots can also be made from various types of metal at a metalsmith's forge as part of both the chain and plate armor set.
High boots provide protection for the lower leg as well as the foot, low boots only protect the feet.
Boots are always made in pairs, but the quality of each is not guaranteed to be matched. Unlike gauntlets, they are generic - no left/right distinction.