I'm not sure if there are any experts on the factions yet, except the devs I guess and they haven't said that much from what I have found. From my own view, I think DeepFried provided the best explanation:
Empire = Feudal totalitarian
Federation = Corporate authoritarian
Alliance = Democratic egalitarian
Although, the political words that DeepFried use are probably not equivalent to "an idiot's guide", which was what the OP requested.
I may be entirely off track, but my own understanding of the factions is this:
- Empire: Futuristic Japan, that went back to their samurai roots
- Federation: Futuristic USA
- Alliance: Futuristic EU
- Independents: Futuristic Norway
Old Japan had a bunch of Shoguns, who ruled supreme in a small area. Similarly, the Senators of the Empire seem to be the rulers of planets. The Shoguns could act entirely in their own self-interest, but the most powerful one would usually get the support of other Shoguns and thus either be appointed emperor or rule over the empire in the name of the emperor. Other Shoguns could build up their power until they could challenge the Shogun that was "favored" by the emperor, so there were massive internal wars now and then during such power shifts (e.g. battle of Shekigahara). I imagine the situation is similar within the Empire; as long as the central Senators hold a lot of power and large armies the Empire is stable, but if a fringe Senator manages to amass power and draw support from other Senators, the whole Empire could become unstable with internal wars being fought between Senator factions. None of that seem to have happened yet in the Elite Empire as of yet, or at least I don't know anyone telling such a story.
USA claims to be the land of the free, where anyone can seize their opportunity and be rich or powerful or both. They survive as a nation entirely on this promise that opportunity is open to anyone. Of course, this isn't true in any way, but it seems the entire populace is so much drawn to the promise of a possibility to be rich/powerful/successful that they don't care about the truth. They hold up a front where it is said that the people vote for one of two parties, a president is elected, and politicians have the power. In reality, companies have the real power and can exert a lot of pressure on the politicians to steer the country in the direction they want to go. In reality, it is a country where the rich and powerful just continue to grow more rich and powerful, while occasionally allowing an upstart to rise to keep up the illusion of "it is possible for anyone". The real power is kept within a central core where no upstarts are allowed in, which sometimes brings up rumours of "Illuminati" or other constructs rumored to rule the entire world. These powerful men will pressure the government to invade countries, or do whatever else they please, in order to increase their own influence over the world. I imagine the Federation to be just like this, but replace country with planets, and replace world with universe.
EU is an alliance formed with the realization that USA, China, India, Brazil, etc are becoming super-powers by sheer population size and economic/cultural influence. Thus, EU tries to allow member nations to keep up their own set of laws and cultures, but also tries to present a unified front towards the super-powers that be. Because several of the member nations already have intimate ties with USA, and because of the sometimes conflicting interests, it is a difficult beast to handle. Similarly, the Alliance in Elite has member planets, all with their own subset of laws, economy, and culture, but they try to coordinate their efforts as a unified front towards the super-powers that be.
Independents, I imagine to be something like Norway. No one is ever completely independent. For example, Norway is a member of NATO and hugely influenced by USA from this membership. But they also have open borders with Sweden, who are members of the EU. The only way Norway survives is because they have oil, and they are willing to trade this "freely" with both USA and EU. Thus, they have a crucial resource and they keep super-powers away from their country by sharing their resource through trade. Compare Iraq/Kuwait, who when Saddam Hussein threatened to keep their resource internally or at least limit trade, they instantly got invaded by both USA and some EU nations to secure their supply of oil. So I imagine independents in the Elite game to follow similar rules; they can stay independent as long as they are not considered to be a nuisance by any super-power.