There are several types of armor based on materials used in manufacturing process.
Metallic armors are usually includes high strength steels, zirconium alloys and sometimes tungsten-base alloys and compounds.
Composite armors which may includes CMCs, MMCs and even CMCs (ceramic/metallic/polymer matrix composites) have been developed recently in order too boost the strength of the materials together with lower weight and costs. Examples:
Al-B4C, Al-Al2O3, SS-Al2O3, SS-B4C, SiC and SiC-base composites
My area of expertise is quite different - ancient military history and technology, but a number of base principles remain the same (until you get into reactive armor in the modern day).
First and foremost it can depend a lot on what you are protecting against. I presume you most likely interested in modern threats. Armor meant to protect against blast damage is different than that meant to protect against ballistics (bullets), and armor designed to protect against shrapnel is different again (and at least in the first half of the 20th century shrapnel was easiest to protect against due to its lower average velocity).
While Behzad's answer is very informative when dealing with modern materials, in the ancient world armor was much more technologically simplistic, and came down "hard outer layer" and "padded under layer." Hard could mean rigid (plate, typically equivalent to mild steel and later medium carbon steel), semi rigid (scale, lamellar, or brigandine consisting of a series of small plates, which could be wood, horn, leather, rawhide, bronze, iron, mild steel or even layered and lacquered paper formed into individual lamellae depending on the culture or time period), or flexible (chain mail, made of interlocking rings, at times secured to a backing but often without, made of iron or mild steel).
The inner layer typically consisted of layered or quilted cloth, or at times cloth and paper that could be made of linen, cotton, or wool (or even horsehair which was used as a filler material). The thickness of this under layer could vary widely, but in general the more rigid the outer layer, the thinner the inner layer. Even when wearing plate armor, you would NOT want to skip some degree of padding, but the thickness would be perhaps 3-6mm. If wearing only a padded layer then thickness could exceed 20mm.
The interesting thing is that depending on what you want to protect against, some surprisingly low tech solutions remain quite viable. Something as simple as paper, if used correctly, can protect against a range of (but not all) small arms fire. It would however require partial replacement of damaged areas after each time it saw actual use.