The soul gem's size determines the strength of the enchantments. If none are available, you first need to disenchant items to learn those enchantments. Select the enchantment and its strenght, where applicable. Different item types accept different enchantments. You can now choose a weapon, armor, or jewelry to enchant. Bring up the menu by activating the enchanter and select the item you'd like to enchant OR disenchant (which teaches you how to do that enchantment). You can find one in whiterun in the castle, Farengar Secret-Fire's room near the throne. Head to an Arcane Enchanter with a soul gem that contains a soul.
Further into the guide, you'll find detailed information filling soul gems, the soul trap spell, and using advanced enchanting in Skyrim.
Disenchanting items goes similarly - you lose the enchanted item to learn the enhancement permanently. You must have a soul gem and know an enchantment.
Here's a quick summary of how to enchant an item. I also cover more advanced topics like how you can boost the strength of your enhancements higher, and provides a full list of enchantments by weapon and armor types. This guide teaches the basics of Enchanting in Skyrim, such as how to enchant an item. Skyrim's Enchanting skill allows you to place magical effects on your weapons and armor. Skyrim Enchanting Guide with Weapons and Armor Enchantment Lists
