Is it Serenity, or Firefly, or...? This point had me confused for a while. Let me spell it out for you. Serenity is the name of the ship used in the show and the film. It's named after Serenity Valley, a famous battle two of the main characters were involved in. Serenity is the name of the pilot (first episode) and the feature film which follows the end of the series. Firefly is the name of the show, and Serenity the ship is a Firefly-class vessel. Clear? If not, see all the things Firefly refers to on Wikipedia.
Anyway, Firefly, as well as the first Serenity, the pilot (but not so much the feature film) is a Western which happens to take place in the future - over 500 years, to be exact. The United States and China pretty much ruled the world in peace, but due to overcrowding, they exiled much (if not all) of the population to the stars, to a new solar system where they found a dozen or so planets, all capable of sustaining life. They terraformed the planets, and tried to start an interstellar parliament, but the outer worlds wanted to stay independent, and a civil war broke out. The Independents lost, and its members generally make up the lower class in the new world (galaxy) order.
Unlike the other space-faring sci-fi shows (Star Trek, Babylon 5, and Battlestar Galactica), Firefly doesn't feature any fancy technology, other than space-faring vessels. Faster-than-light travel must exist, but it's never discussed. At all. Ships don't have weapons to shoot each other with, or at least Serenity doesn't (it's a cargo ship). No teleporters whatsoever. Guns fire bullets. There are a couple laser guns, but they're fancy and rare. The bad guys are kind of like the Empire in Star Wars, and the good guys are outlaws. Kind of like in Star Wars, I guess, but without lasers. And no magic, either, although they do have a latent telepath, and they do have some pretty impressive fighting, though mostly only in the feature film.
The Western element shows in how people dress and act, at least on the outer worlds. And the sort of code of honor among the band of outlaws. A newcomer to the crew tells the captain (a real cowboy) he's afraid he'll shoot him in his sleep. The captain says to him, "If I shoot you, you'll be awake, you'll have a gun in your hand, and you'll be facing me." Scenes and lines like this define most of the characters, at least the main crew. In the pilot episode, the crew picks up a priest who is more than he lets on (though they never fully explain him) and a pair of relatively young siblings, a genius doctor who rescued his sister from a research lab, and now they're both on the run.
The show only lasted for one season, but was followed by a good feature film, and, though short, is definitely worth watching. A couple issues are left unresolved, but it isn't anything major; they wrapped up all they wanted to wrap up, being aware the show was being canceled before they finished making it, so anything left open by the show is mostly answered in the movie.