Smithing is a powerful and enjoyable way to form metal ingots into tools.
To start smithing, you must first add fuel to the forge by holding shift + right mouse click, then add your ingot in the same way. To ignite the forge, shift + right click it with a torch. Once the metal is over 60% its melting point it can be worked. Take it from the forge (right click) and place it on the anvil (shift + right click). This will open up a dialog allowing you to choose what to tool/weapon to craft.
Your goal is now to fill in the empty blue squares with metal. Hit 'F' with a hammer in your hand to see your tool modes. The first mode spreads metal in all directions, the next 4 in a single direction and the last one removes a piece of metal. If your work item cools down too much you will need to reheat it on the forge.
If you run out of metal while forging an item you will need to add another ingot. Place another hot ingot on top of the work item for refilling.
Here is a video of a complete smithing process