It seems that emperors and their wives, as well as commoners, senators and everyone in between had connections to these groups, which were based around chariot teams and were identified by color.
What kind of social power did they hold?
I talked a bit about their political roles in my answer here. The circus factions were important because they were one of the few ways for the common people's voice to be heard by the emperor, as the emperor attended the games and heard the crowds' acclamations in an almost ritualistic fashion. In the sixth century, it on occasion literally decided who would be emperor, since although the emperor was essentially an autocrat, his popularity was still important. On a less political level, the circus factions were just like professional sport teams and their fans today, wielding huge amount of power and influence over both the rich and the poor.
The Nika Riot was one manifestation of this influence, since as soon the two factions decided they had a common grievance (that the emperor was unjust in executing a Blue and a Green supporter), they ignited a riot that consumed Constantinople, leading the death of thousands and almost ending Justinian's reign. However, there is evidence that Justinian almost deliberately let the riot run its course because either he was unwilling to attack the rioters at first or used it as a ploy to weed out his opponents amongst the upper classes. Procopius is not the only source we have and according to Marcellinus Comes, an older account, the riot from the start was a power-play by senators who 'urged' the crowds to continue their rampage as a cover for an usurpation attempt. Then again, this may just be the propaganda released by Justinian's regime to justify purging the senators. Basically, because we have several sometimes conflicting sources describing this event, you can either view the Riot as one driven from below by popular discontent over Justinian's officials/policies, or one that was manipulated by or hijacked by upper class interests. It all depends on the personal inclinations of the historian to be honest.
Also, I'd just like to add that we don't have enough information to ascribe an 'ideology' or theological position to either the Blues and the Greens, so it is better to view them as a factions with fluid ideals and goals. The idea that one faction was more conservative and the other more progressive is an outdated point of view :)
The Byzantine Emperor Justinian was a supporter of the Blue team, which supported the largest, oldest and most successful of the chariot racing organizations in Constantinople. He was basicly like a fan of the New York Yankees or Ferrari F1 team. For much of the history of Chariot racing, there were other fan groups that supported smaller stables, but during Justinian's reign, they had coalesced into one large faction, known as the greens. There are complex theological divides and widespread dissatisfaction with Justinian's policies in many areas. High taxation was the biggest of these discontents.
Procopius provides a surviving account of the Nika Riot that broke out in 532. http://www.fordham.edu/halsall/source/procop-wars1.asp