Armor is traditionally constructed from leather and steel, with hempen cloth, brocade, damask or silk for fabrics and laces. Silverstari armour is generally designed to be as light as possible, providing for ease of movement but compromised protection (this of course varies from kingdom to kingdom. The Kingdom of Yu tend towards heavier, bulkier armour). Light armour is favoured as samurai engage in hand to hand combat where fast and precise movements are required.
Great armour is called Ô-Yoroi (the Ô prefix meaning "large"), and generally refers to old-fashioned, boxy armour.
Gusoku is the term for a full suit of armour.
Pieces of Armour
Menpo (face mask);
Do (cuirass) for the torso (usually has a mon on prominent display);
Sode (shoulder guards);
One or two kote (armoured sleeves) for the arms;
Tekkô (hand guards);
Kusazuri (armoured skirt or apron) to protect the upper thighs;
Suneate (shin guards);
Haidate (thigh guards).
Kôgake (armoured tabi (shoes)) are optional as foot protection, but not good on the feet. Kegutsu, fur boots, are better for mobility.
A sashimono, while not a piece of armour in itself, was often added as an identification aid for warriors in pitched battle.
The uwa obi is the sash through which one would wear swords, fans, etc.
The lacing of the armour is called Odoshi, and is made of dyed leather or silk braid. Inexpensive armour used cotton.
Under the Armour
Lower ranked samurai would wear their day-to-day clothes beneath their armour. Typical samurai would wear hakama (trousers) and a shitagi (a form of kimono with a single button at the throat). The more prestigious would wear an elaborate hitatare (robe which includes hakama) over their kimono.
Putting the Armour On
The convention is that one puts on arrmour from the bottom up, left to right. The exception is that armoured foot gear — as opposed to normal straw sandals — must be donned after the suneate are on. After the kabuto, the obi and sashimono can be worn.
Armour could be stored in wooden chests called gusoku bitsu. There is actually an official way of putting armour away, including the order one puts the various pieces inside and how to arrange what goes where.
(I highly recommend this link for armour purposes. It has pictures and details of various armour types and is very informative. It includes entire sections on how to put armour on, store it, and make it.)