So, in Firefly, the Alliance controls all of the planets in the system. But I don't think it is ever explained in much detail how that works. Is it a democracy of some kind, or something else entirely? How does the Alliance govern its people, and how are the leaders of said alliance determined?
The Alliance is a totalitarian-appearing government, controlled by a Parliament and a concordance of worlds who maintain control over those worlds using a rigid, authoritarian government linked to an oppressive military force. It is the sole government of the Firefly universe, and was the winner in the Unification War against the Browncoats, spoken about regularly during the show.
In Serenity, the movie set in the Firefly universe, it is revealed that the Alliance is run by a branch of the government called the Parliament. --Wikipedia Entry > Alliance (Firefly)
The two main worlds of the Alliance are Londinium and Sihnon. The Alliance supposedly leads a multi-racial, multicultural and multi-religious universe. It has strong overtones of Corpocracy, since advertising and media are pervasive and mostly done by the omnipresent "Blue Sun Corporation."
In any large bureaucracy, there will be a failure to communicate. At this writing, (11/2013) there is a scandal in the USA about the NSA which apparently, during the GW Bush administration, began recording the phone calls and emails of friendly foreign governments and elected officials. This continued into the Obama administration, without the senior political levels being made aware of the practice. (Perhaps just a case of "don't ask, don't tell").
Never put down to conspiracy what can be explained by stupidity, laziness and incompetence.
The Alliance is so-called because it was formed from an alliance of the US democratic government and culture and the Chinese one-party government and authoritarian culture. Even today, large businesses are strong voices in both cultures.
The Alliance was formed originally on Earth-That-Was to coordinate the exodus.
In practice, although the Parliament seems to be an elected body, we have seen that there are slave-holding societies and indentured worker societies (Higgin's Moon) . Some worlds, like Persephone (Shindig) have an aristocracy. For American readers, yes, it is possible to have an aristocracy and still be democratic, for example the United Kingdom and most of Scandanavia are both. It is possible that not every world allows universal suffrage. It is possible that each world sets its own standards for voters.
It seems likely that the "rigid authoritarian government" and "oppressive military force" may be less thorough in their control than the Browncoats imagine/believe. Again, in Train Job, the medicine for the miners is being sent by the central government. There is no mention that this medicine will only be available to those who can pay for it. And in the same episode, the theft is contracted to Serenity by a criminal who seems to act with impunity -- he even has his own space station (War Stories)-- in spite of the chaos uncontrolled criminality causes in society. A rigid government with an oppressive military force available would be shutting Niska down, well before he gets to such control.
Blue Sun Corporation is a vast and almost monopolistic organization. Again, in any large bureaucracy there are going to be outliers who have their own projects, unnoticed by the central authority. Some will be benign, even useful to the company and the general public, others will be dangerous.
It seems likely from what has been said by Whedon and in the various books and commentaries, that The Academy where River was subject to experiment/torture, was not technically a Government facility but a Blue Sun facility, likely under contract to some part of the Government. In the BDM, the Operative is upset that Dr. Mathias has allowed River, a psychic, in the same room as senior officials with grave and dangerous secrets, so there is definitely a cross-over.