This article contains one or more spoilers. This is a potentially bad thing. Be careful reading on if you are not well-versed in the continuum or not caught up!
Firefly was a short-lived (one season before being pulled by Fox for "low ratings") television program centered on a group of misfits flying around the galaxy (or "The 'Verse") in a beaten-up Firefly-class spaceship known as Serenity.
In this version of the future, Earth has been abandoned in favor of a new solar system with dozens of terraformed planets. The Alliance, a Sino-American amalgamated superpower, dominates the politics of this system and governs all of the Central Planets, and, technically, the outer planets too.
A recent civil war was fought between the Alliance and the "Browncoats," who were anti-Unification, the name given to the amalgamation of the government of all the planets of the system.
The Fireflyverse (including the film adaptation/continuation, Serenity) is considered to be part of the Jossverse, named after its creator, Joss Whedon.
Captain Malcolm "Mal" Reynolds
The captain of Serenity (played by Nathan Fillion), who was a sergeant in the Unification War. Little of his past is revealed before his role in the war save that he grew up on a ranch on the planet Shadow, with his mother and an unspecified number of ranch-hands ("about forty"). No father is mentioned.
Mal strongly opposed and continues to oppose Unification and the Alliance. He believes in people governing themselves, rather than an impersonal galactic government that steamrollers the little people. He is ferociously loyal but can be conniving and underhand in his dealings to see that he and his get their fair share or more. His attitude to the law is flexible, and he frequently steals and lies, and to a lesser extent kills, without too much bother from his conscience. His main love is his ship, Serenity, which is his home, despite her general unkempt air and lack of maintenance.
Zoe Alleyne Washburn
Zoe (Gina Torres) was a soldier who served under Mal in the Unification War, and when the war was over she followed him to Serenity and a life as a smuggler/transporter of goods. She also found a husband thanks to Mal, marrying Hoban "Wash" Washburn, a pilot he'd hired. Zoe is by turns taciturn and loving, a stoic warrior-type, loyal to both her captain and her husband (illustrated best in the episode "War Stories"). Zoe is Mal's second-in-command on board Serenity, and a crack-shot with her mare's-leg style sawn-off shotgun.
Hoban "Wash" Washburn
Wash (Alan Tudyk) is the pilot of Serenity, hired by Mal against Zoe's wishes (in flashbacks she is seen to warn Mal that "there's just something off about him"). Nevertheless, he and Zoe fall deeply in love and get married. He is an extremely fine pilot, and also a joker and all-round comic relief. He keeps little plastic dinosaur models on his pilot's console (no relation to an agent's console, although there are a few agents known to keep plastic dinosaurs there) and plays with them in his off-time. At one point he suffers a concussion at the hands of a villain whom he refuses to be seduced by, waffling on about his love for Zoe until the frustrated con-woman is forced to kick him in the head. In the film, he is killed by a Reaver javelin hitting him in the chest.
An AU Wash was recruited to the PPC sometime in 2008. The exact circumstances of this are as yet unknown, but he has been assigned to the Department of Sufficiently Advanced Technology as a pilot and mechanic.
Kaywinnet Lee "Kaylee" Frye
Kaylee (Jewel Staite) is the young genius mechanic, hired by Mal when she fixed Serenity's engine after having noticed the problem while flat on her back banging the previous mechanic in the engine room. Said previous mechanic was fired and Kaylee hired, as soon as she'd done her dress up and run off to ask her parents. Kaylee is young, chirpy, and quite girly; while delighting in practicality, engines, grease, etc., she still adores pretty clothes and the attention of young men; specifically, Simon Tam. Kaylee seems to have almost a brother-sister relationship with Jayne Cobb, and to be almost a daughter to Mal.
Jayne (Adam Baldwin) is a mercenary, not well-educated (although he can read after a fashion), and determined to do whatever it takes to get the best profit from a situation; an attitude that occasionally puts him at loggerheads with Mal, especially when a reward is offered for the capture of Simon and River Tam. He loves weaponry, his favorite being a large gun named Vera. Jayne joined the crew when his former gang were attempting to rob Zoe and Mal. Mal asked him how much he would get from the takings, and offered him a better deal, whereupon Jayne shot his former boss in the foot and changed sides.
Inara (Morena Baccarin) is a Companion, a sort of prostitute, but being a Companion is a highly regarded and much-honored profession. She is calm and composed, very well trained at reading body language, speaking several languages, playing the dulcimer, and many other civilized arts. She rents one of Serenity's two shuttles from Mal, and her presence on the ship gives them a veneer of credibility when they visit Alliance-controlled Core planets. There is a tremendous amount of UST between her and Mal. Despite being respectable, and not being a member of the crew, Inara does on occasion save the day.
Shepherd Derrial Book
A Shepherd is essentially a priest, and Shepherd Book (Ron Glass), after leaving his monastery, decides to take passage on a ship and spread the Good Word amongst the peoples of the galaxy. Unfortunately, he chooses Serenity, where Mal is hostile to his God and gets them involved in crimes almost every day. However, Book rises to this challenge admirably, displaying a worryingly thorough knowledge of criminal procedures, government procedures, weaponry, and tactics. Sadly, we never find out his background, and he remains one of the mysteries that could have been solved if only Fox hadn't been stupid and pulled the series. Like Wash, he is killed in the film, dying from wounds sustained during the Operative's attack on the town he was living in. Before being mortally wounded, however, he personally shot down the ship the Operative sent to attack his town.
Dr. Simon Tam
Simon (Sean Maher) is a first-rate doctor, trained on the Core planets, and coming from a rich family. He is extremely smart and very well educated. He has seldom had to do manual labor in his life, and cannot fire a gun or really be useful in combat, which are the main reasons that Jayne and he have a vendetta; Simon's view of Jayne is "you're like a trained ape! Without the training!" Simon's main goal is to protect his sister, River, whom he sprung from a secret governmental institute where they were experimenting on her brain to an unknown end. Both siblings are fugitives from the law. He has strong feelings for Kaylee but is unable to express them, as she seems to render him tongue-tied and only capable of spouting extremely bad-taste jokes that eventually offend her.
River (Summer Glau) is Simon's younger sister. She is a certified genius, and appears also to be psychic, and possibly schizophrenic after her ordeal in the secret government facility that Simon rescued her from. She and Kaylee are close friends, but she is erratic and prone to violent behavior, weird comments ("also, I can kill you with my brain"), etc., and sometimes worries people deeply because of this. Simon is attempting to cure her. However, her strange abilities have, on several occasions, served to extricate the crew from a tricky spot.