Okay, more details. . .
This is supposed to take place in our galaxy, but about 1500 years in the future. The setting is a planet, Pathros, on which one will find dragons. Now, the dragons that the dragon-riders ride are the classic dragon type, but the other species of dragon on Pathros are not. All native creatures of Pathros are considered dragons, because they all have some trait about them related to the elements. For some, it is the fire breathing, or the gas breathing, and so on, but for most, it is different. For example, one species of 'dragon' I've put into the story is the pathrosic constrictor, a massive snake-like creature, whose very body emanates frost, leaving a trail of frozen ground behind it as it slithers along, or freezing its prey as it coils around them. Even the plantlife on Pathros has similar qualities, like the hearthoaks, deciduous trees whose pod-like seeds erupt in fire if touched. So, the 'dragons' could be anywhere from small, electrically charged rodents, to massive beasts with fire running down their backs.
For dragon riders, however, the species they ride look just like the dragons we generally think of today.
Now, the function of the dragon riders is tied to the way the government of Pathros is set up. The planet would appear to be ruled by several city-state-like Houses. The Ruling Houses, however, are not actually governments, but businesses. Pathros is, in reality, governed by an Empire that is based on an entirely different planet. The only reason anyone is on Pathros is to harvest Aiex, an extremely valuable chemical which is a large part of the story. So, any city on Pathros is only there to house those who harvest the Aiex. With that in mind, all cities are owned by the numerous businesses, or Houses, who run Pathros. I hope that made sense.
Each House has their own force of dragon-riders. some houses have as few as a dozen, while the largest of them has around 150. Officially, they are merely a police force. However, they are secretly used in the constant war going on between the Houses. Everyone is perfectly aware of how the dragon riders are used, but everyone pretends otherwise.
Another important note is that the dragon-riders are among the few whose bodies can survive an injection of aiex, which will greatly increase their capabilities. This, and the fact that they ride around on the backs of dragons, make them the greatest warriors in the galaxy.
I hope thats enough info to get some ideas, and I hope that I presented it in a comprehensible way. Looking back through this post, it seems to be a bit jumbled.