I believe that the type of caste system used by each race depends largely on what kind of creature they are. Presumably for races much like us, self-centered, selfish, and empathetic for other members of the same race, you would have governments much like ours:
Each system of government implies a little bit about its people:
Leader of this race rules with an iron fist. He likely only obtained that position because he was the strongest or because his father was strongest. Leaders in this type of system, at least with large groups of individuals, don't tend to last long so you wouldn't see long lines of leaders stemming from the same bloodline, because someone somewhere down the line would have contested the current rule.
People under this rule are simply told what to do, and they are expected to carry it out. There is no sense of nobility or purpose to what they do, though some may still blindingly defend their way of life because they see it as characteristic of their species and to change this would mean changing who they are as a species (take note, good plot points in this). The modern-day equivalent of depotism is a dictator. Dictator gets all the spoils, and his people none. The people exist only to uplift the dictator in his glory.
Referring to lord of the rings, I would expect to see races like the Uruk-hai or Orcs to follow a depotism type system.
Monarchy, like depotism, has a single ruler that has absolute control over his people. However, unlike depotism, the people follow him because he is the chosen leader, likely even religiously. He may not necessarily be the strongest, but he is the one most people rally behind. Not everyone will support him, but with a carefully balanced caste system, everyone knows his/her place and doesn't rebel. Too much imbalance in the system will lead to open revolt, though a revolution under a monarchy tends to lead to another king, not a new form of government. However, most people only wish to serve their lord well, and take pride in their banner.
The king does not make all decisions. He has large tracts of land that he gives to lords, who in turn replenish the treasury. Those lords are like kings of their own land, who in turn delegate their tasks to lesser lords until you arrive at the lowly serf, who doesn't own the land he works on, but is able to provide for his family and live off the land in exchange for paying a tax.
Contrarily, you'd have nobility, and courts, where the king would host extravagant parties. Relationship with foreign entities is very important to a monarchy because image is very important to a monarchy.
What this says about its people is that almost in contrast, people do not care about what happens to the other guy, and yet, they have no problems rallying for a cause or an idea. It is enough to have hope and stability.
Again using lord of the rings as a reference, a race that would use a monarchy would be the dwarves. Perhaps they would not farm land, but they would mine coal, however, the system would be very much the same.
Those under a republic would be slightly enlightened with respect to a monarchy or depotism in that a single ruler cannot be trusted. A republic embraces this idea and decides that an idea is worthwhile only if it is an idea shared among several leaders, not just the one.
The people under this type of government would be free to pursue any type of living they want, however, one cannot survive in cities unless he or she has something to offer, so you would likely see a strong tendency towards trade. Unlike under a monarchy system where your role was determined before you were born, under a republic, you can make vases if you are talented at making vases. As such, the arts would flourish and items of true value would be available, made by those who have perfected the art over many generations. Alternatively, you can even pursue a life in law and study and become a well-respected member of society, of which select few even become senators or council members. While there would almost surely be an elitist mentality, nobody is purposefully kept down in this society.
Races under monarchies would have open trade with races under a republic, however they might secretly fear how people might be pulled towards this type of government, and therefore despite the outwards appearance, there might be much resentment underneath.
The race that I would expect to see under a republic would be the elves.
You may not think that Democracy would be possible, but then, mankind has democracy so it must be possible. This is probably on the higher-end scale of the nobility of systems of government, and something that not even we were able to accomplish without lots of trial and error in history. Keep in mind that this isn't to say that everyone cares for their fellow man in a democracy. You still see politics, corruption, and illusion of choice in modern-day democracies, and I would expect nothing different in other races.
However just the same, it takes an enlightened people to see the need to adopt democracy. You don't often see this in fantasy books, because democracy is a relatively modern form of government, and most fantasy books tend to take the 13th century as a frame of reference for technology and culture. This doesn't mean that a democracy couldn't exist, however.
This might imply that the people under a democracy are much more prone to care about one of the same race. This doesn't necessarily mean that they would care about someone of a different race, perhaps rather showing disdain instead.
You may see a group of senators or a single leader perform decisions on what to bring to a vote, and while there may be corruption or trickery, it would be well-hidden.
Another idea of a system of government that obviously wouldn't be applicable to human beings might be a hive-mind. Each individual would be capable of telepathy. There would likely be an interesting contrast between each individual having a voice and each individual being a drone at the same time. There would be very little protest to decisions being made because in a certain sense, every individual is a piece of a larger whole. If you were to talk to one such individual, for important enough discussions, it would be like having made a speech to all of them, with their ease of transmitting ideas.
There may or may not be a central leader that acts as a tie-breaker. This central leader could be simply another member of the species or it could be a queen-like monstrocity. The only question would be how much influence this queen would have on the rest of them. However, whether it be her decision or a collective-decision, you would still see mostly consent between all of them.
It would likely be difficult if not downright impossible to get them to use "I" in the place of "we", seeing as they would likely have difficulty distinguishing one another. When they do set their minds towards doing something, they are relentless and unstoppable.
Other races would likely steer clear of this race, though it might happen that they occasionally need to cross paths.
An example of a hive mind system that comes to mind is the Borg in star trek.