Stars, or stardust, are rather like experience points in traditional MMOs. Most jobs earn stardust by playing minigames or engaging in combat. Adventurers earn stars by completing collections or special quests. The stardust that you gain only counts toward the job that you are currently in.
For all jobs, these stars are simply used to help the job level, effective on 12/14/2009.
|Allakhazam credits this post by many people at the SOE Official Forums for some of the info in this article.