The only other survivor of Malcolm Reynolds' platoon from the Battle of Serenity Valley (it is assumed that Tracey? was transferred to another outfit after the Battle of Du-Khang, as mentioned in The Message), Zoe joined him as Serenity's first mate to kick a little ass around the universe. And after a hard day committing mayhem wherever possible/necessary, she gets to come home to her husband Wash, Serenity's pilot.
Zoe is known to be a sensible "warrior woman" with a dry sense of humor. Mal always defers to her on important decisions. Her past remains a mystery, but it is implied that she spent her childhood on a ship, hence her fondness of flying. This could have been the commonality that brought Wash and Zoe together, despite her initial dismissal of him (likely because of his mustache).
Her relationship with Wash humanizes her. It allows the audience to see that Zoe can encompass multiple roles and still not be diminished any less by any one of them. The fact that her ability to fight does not encumber her femininity is especially noteworthy in the world of genre writing. Zoe's an excellent wife and soldier.
Her relationship with Mal is also special in that they are close without ever having been in a romantic relationship. That's another rarity in genre writing. There is an easy tendency to "couple" close male/female relationships. But Mal and Zoe's relationship is probably closer than many romantic relationships. They were soldiers in the same company, they only two who survived it. The relationships of soldiers who have served together is close-knit, indeed.
Zoe's relationship with Mal is complicated, however, in that she must manage the delicate balance of deferring to both Mal and her husband Wash. *see WarStories
Zoe's role is an important one in that she sets the tone for rest of the crew to follow Mal's leadership. She chooses to address him as "Sir", despite the fact that the war and her official service under Mal is over. So, when others in the crew challenge Mal's leadership, as often occurs, Zoe's respectful address and always ready back-up, helps maintain the chain of command.
Of all his crew-members Zoe arguably knows Mal best. Zoe is the only crewmember who knew Mal before he lost his faith. *see Serenity Pilot
Inara, however, seems to have motivations most similar to Malcolm Reynolds'.
According to Serenity: The Official Visual Companion, by Joss Whedon, Zoe's last name is Alleyne.