Note! This mission is optional and can appear in your journal under various, random names depending on the objectives. I'd recommend playing it over from time to time, as it's an easy way to earn gold. You will get the chance to receive this mission no sooner than after finishing the first main College of Winterhold quest, First Lessons.
Find Sergius Turrianus (usually he's walking along the main square or at the Hall of the Elements) and ask him if he needs your help. One of the missions that he gives you might be heading to a certain person who asked him to enchant an item. Leave the College of Winterhold, open the world map and head to the random location. Approach the client and ask him for the random item which needs to be enchanted (screen above). Afterwards return to Sergius to receive a prize in gold.