Leather Armour is created by using the pelts of animals that you have killed. By using the tanning rack at most blacksmith workstations, you can turn pelts into leather, or turn leather into leather strips. Leather is needed to smith or improve leather armour while leather strips are needed to create all types of armour.
Iron Armour is created by using the previously mentioned leather strips in combination with Iron Ingots at the blacksmiths' forges. Iron Ore is required to create Iron Ingots. Iron ore can be mined from ore veins scattered throughout the rocky landscape of Skyrim through use of a pickaxe. Iron ore in your inventory can be refined at a smelter (such as the one at the Whiterun blacksmith) to be transformed into Iron Ingots.
Steel Armour requires Steel Ingots in addition to Iron Ingots and Leather Strips, In addition you will need to have invested in the base perk of the smithing skill tree. Steel ingots are created by smelting Iron Ore with Corundum Ore in a smelter.
Elven Armour requires the Elven Smithing perk in the smithing skill tree. The base material for all Elven armour is Refined Moonstone which is also used to improve existing weapons and armour. To craft the gilded armour you must also have Quicksilver Ingots.
Dwarven Armour requires the Dwarven Smithing perk in the smithing skill tree, which additionally requires at least 30 proficiency in smithing. In order to get the required materials, you will need to track down broken dwarven artefacts to smelt into Ingots; a good place to search for these are the various Dwemer Ruins scattered near the Morrowind border (eastern Skyrim).
Glass Armour requires Refined Malachite, a green and blue ingot, to smith and improve.
Ebony Armour requires Ebony Ingots and the Ebony Smithing perk (80). Ebony can be found in mines close to Orcish strongholds. Ebony is the only armour that requires only Ebony ingots to make.
Deadric Armour requiring ebony and a Daedra Heart, this armour can be crafted after the level 90 smithing perk is added.
Dragon Armour: There are two types of dragon armour: Dragonplate and Dragonscale. Dragonplate is a heavy armour crafted from Dragon Bones and Dragon Scales. Dragonscale is a light armour crafted from dragon scales.
You can pay Eorlund Gray-Mane at the Skyforge to train you in smithing after joining the Companions, assuming you have enough gold at your disposal.
Having difficulty acquiring raw materials? Enemies occasionally carry metal ores or Ingots.
Some areas such as the summit of the Throat of the World or various mines scattered about Skyrim contain large concentrations of ore veins.
You can purchase a selection of Ingots and leather from any smith.
Jewellery is another good way to quickly increase your Smithing skill. Combined with the Transmute spell to quickly obtain gold and silver and crafting rings for double the effect earns more experience per unit of iron ore than producing iron goods and requires no additional ingredients.
You can buy a whole stock of iron ingots and leather strips, (or acquire them by other means) to make lots of iron daggers to quickly raise the smithing skill and grow a few levels at the same time. When you have finished, you can sell the daggers to get some of your money back. If you enchant them before sale, you can also increase your enchanting skill quickly while making more money from an enchanted dagger.
Improving Armour and Weapons
Grindstones are used to improve weapons - You can sharpen your weapons for extra damage. This requires Ingots and a maximum of two upgrades can be applied to a weapon with the applicable perk unlocked.
Workbenches are used to improve armour - You can upgrade armour for extra armour rating. This requires ingots or leather. Armour can be upgraded to six levels of quality with the applicable perk unlocked. In order, these levels are: Fine, Superior, Exquisite, Flawless, Epic and Legendary.