River Tam and her brother Simon live on a Firefly-class spaceship named Serenity approximately 500 years in the future, when the series and film are set. River was considered a child prodigy when it came to almost everything, smart beyond her years (she was correcting her brother's spelling when she was three years old). Her gifts were not merely intellectual; she is also a talented and graceful dancer.
When River was about 14, she was sent to a government learning facility known only as "The Academy". While her parents and Simon believed the Academy was a private school meant to nurture the gifts of the most talented children in the Alliance (the uniting governmental force over all inhabited galactic planets), it was in fact a cover for a government experiment in creating the perfect assassin. While in the clutches of the Alliance doctors and scientists, River was tortured and experimented upon secretly and extensively, and made the victim of cruel surgeries and psychological abuses.
The Alliance attempted to isolate River from her family, but she managed to send a call for help by putting a coded message in a letter to her family. Simon decoded the message, and set out to rescue his sister, despite his parents' insistence that he was simply being paranoid. Simon exhausted his own personal fortune and sacrificed a promising career in medicine, but eventually located River with the help of anti-governmental groups.
Rescued by her brother and put temporarily in cryo suspended animation, the two became wanted fugitives, were disowned by their parents and hunted by both Alliance forces and bounty hunters, who have been enticed by a large bounty. Simon, a talented doctor, treats her regularly, trying to find medicines that help the now-mostly psychotic River. He discovered through an advanced form of medical imaging that part of her brain, the amygdala, which acts as a filter for emotions, had been stripped. This prevents River from being able to control her emotions, even if she wants to. As Simon describes it, "She feels everything; she can't not." ("Ariel").
River displays near-telepathic perception: she "sees" and foretells the future (albeit often in a twisted, nursery-rhyme sort of way), she can "feel" people's emotions, sense their thoughts and knows what is happening in places beyond her sight (e.g. telling Captain Reynolds "don't make faces" through the intercom while in a separate spaceship). What River sees is open to interpretation, although it often is very clearly related in some way to a future event (or, in some cases, events she simply could not know, like when she insinuates that the Patron of a small village came to power by killing the previous Patron, or that she knows of crewmember Jayne's betrayal). At the end of "Trash", she threatens Jayne by very casually remarking "I can kill you with my brain", but it is unknown whether this was an empty threat or not.
She is deeply intuitive and relies very much on feelings of things rather than knowledge (for example, after Serenity transported a herd of cattle between worlds, River tells Mal that, while in transit, the cows "forgot how to be [cows], but then they saw the sky, and they remembered again") and her way of communication is almost lyrical, always deeper than it appears and never simple (save for a few occasions, e.g. telling Simon "You're such a boob"). This superhuman insight may come from some psychic talents, or may be a manifestation of her sheer genius and ability to process all the information regarding a situation (including body language, vocal inflection, prior knowledge and anything else that may be relevant), much in the way that Sherlock Holmes would often make brilliant deductions about total strangers. Her abuse at the hands of the Alliance may also have helped to augment these talents.
The full extent of her psychic/intellectual powers are currently unknown, as are the intentions of the shadowy organization who experimented on her, or even the true relationship that organization has to the Alliance. River has demonstrated superior skills in fighting and weapon-handling in several circumstances. In "War Stories", she shoots three soldiers with her eyes closed. In the film Serenity, she takes out an entire bar full of dangerous patrons and security personnel, and ultimately confronts and destroys a mob of Reavers single-handedly.
By the end of the film, River appears to have stabilized, suggesting that much of her disassociative behavior was related to the secret she revealed about the nature of the Reavers. Her extreme genius seems to be intact, however. (In a scene edited from the final cut of the film Serenity, River calculates the most efficient escape vector and launch window for the ship to leave the planet Miranda… entirely in her head, apparently without paying attention to the intensely complicated mathematics). At the end of the film, she takes over the co-pilot's station and capably flies the ship (under the watchful eye of Mal).
In the R. Tam sessions, a webcast series of short films produced as tie-ins to the movie Serenity, some of what was done to River is shown, including the various levels of insanity. We get to see River when she first joins the Academy. The interviewer asks her what she "sees" and she tells him about how their first experiment died on the table, hinting even more at her psychic powers. She later kills the interviewer with a pen.
Disclaimer, I am neither River Tam nor Summer Glau. I am just borrowing them for fic and rp purposes. This is not real, this is for Roleplaying purposes only.