The Serenity Role-Playing Game is an officially licensed System and Setting set in the 'verse of Serenity (and the more popular Firefly TV show). The game is published by Margaret Weis Productions. Alternatively, you could just play Traveller. Still, the book does give some good setting notes. The basic idea follows the premise of the show, for the most part. The players are the crew of a ship, trying to make a living (honest or not) by taking various jobs, hauling cargo, getting in firefights, running from the law / reavers / what have you. A creative GM might come up with some interesting stories, but plenty of fun can be had just with the basic concept alone.
The core system is the first version of the Cortex System, which MWP reused later on a bunch of licensed games, and then brought out as a standalone general purpose system. All core attributes are measure in dice sizes, so a weakling might have a strength of d2, and a strongman might have a strength of d10. Skills are also measured in dice, and actions are checked by rolling your relevant skill + relevant attribute, adding the dice results, and checking against a target number. If this sounds weird to you, well, there's a reason we're not all playing Cortex these days.
Firefly: The Game - A resource gathering competitive board game based off the same setting. Each player has their own ship and follows roughly the same sort of tasks that the RPG presents as default (Take jobs, Misbehave, Get paid, Keep Flyin')