Body armor probably started out as a bunch of leather or cord; some people used cloth as an armor. The Saracens wore silk shirts, because while arrows would pierce the skin, they wouldn't cut through the shirt, so you could pull out the arrow by pulling out the shirt.
There is something made of multiple layers of linen, which I think is a gambesan, though I know that's not the correct spelling. This was apparently a very effective armor in and of itself, and I know for certain that it saw use in Europe at some point. Highly likely that this was worn under other types of armor as time progressed.
Leather has a variety of uses in armor. Vegetable-tanned leather, when boiled for at least a few minutes, gets hard as a rock after it dries. While this can be used to make plate armor, I don't know if it was ever used as such - its use might have been restricted to helmets, greaves, breastplates, and other large, easy-to-make pieces.
Another way to harden veg-tan leather is to put it in a pan with beeswax in your oven at about 225 Farenheit and fiddle with it until it is coated in the wax. Be sure to soak and shape the leather before you do this.
Leather is also used to keep the numerous metal plates that make up plate armor in functioning order, and on their wearer. I have yet to see any piece of plate armor that is not secured by leather straps (except greaves - some of them fully encircle the lower leg in metal, and have metal hinges). Some people don't realize that the flexible joints and such use leather hinges, as opposed to rotating on a rivet of some sort. Say you have something like a knee joint of plate. Down both edges on the inside, a leather strap is riveted to each individual piece in such a way that the pieces can move about each other properly.
Eventually, people starting making armor out of bronze. I've seen a picture of some oddly-jointed Greek bronze armor that was kind of like primitave full plate. Mostly, they just made breastplates and greaves, and also the large round shields that the Hoplites carried.
Another type of armor is lamellar (spelling?), which is vaugely like scale mail, except people used horn and hardened leather in addition to metal.
I'm not sure when chainmail was invented, but it got stronger and stronger through the ages (progressively thicker gauges of wire, and progressively smaller diameters of links). Eventually people started putting steel plates over the mail, and from that you got full plate, which most people didn't use anyways, because it was hideously expensive.
Firearms hearalded the end of metal body armor.