Princess Luna

Depictions
Prior to Season 2 episode 4, fans had nothing to base Luna's character on except for a couple lines in Season 1 episode 2 where she apologizes to Celestia for her Nightmare Moon tantrum. This led to a variety of guesses as to who this smaller alicorn really was and why she was apparently too busy to visit Ponyville again during Season 1. The following are some of the more common or popular character traits:

Studying
Luna had been locked in the moon for 1000 years, and Equestrian culture had marched on without her. As a result, Luna needs to spend most of her free time reading up on history, law, and other subjects that have been significantly changed over the past millennium.

Luna's behavior in Luna Eclipsed basically jossed this idea.

Socially Awkward
Again, 1000 years of exile isn't conducive to being able to interact with others. While few were able to correctly predict Luna's canon behavior, many fans rallied around the idea that Luna had become shy and unsure of herself and needed assistance, whether from the Mane 6 or members of the castle staff, to learn how to step out and take risks.

For better or worse, this seems to be a dominant part of Luna's canon characterization.

Trolluna
In more light-hearted depictions, Luna is seen as being a prankster in a similar vein to her sister Trollestia. The sisters may get into prank wars that send the staff and guards running for cover, or conspire to lighten the boring court sessions with strange happenings.

Reluctant Co-regent
For one reason or another, Luna does not feel fit to take back her role in raising the moon and managing the night-time affairs of Canterlot, or feels that her contributions aren't given the same weight as her sister's. Celestia will either give her an encouraging speech and opportunities to take charge of smaller things, or simply declare a need for a vacation and drop her sister into the throne for a week.

Confident
This Luna has gotten over most of the big issues and has the gumption to take matters into her own hooves to solve the minor ones. She's also confident enough to take over Celestia's job in times of crisis, or simply because she wants it. In a negative light, this trait equates to Luna remaining jealous of Celestia and being willing to stage another coup.

Gallery

 * See also a gallery of assorted Luna images