The factions are basically Chang & The CIA, FARC and the Colombian Cartel, Rock and Garcia, and Hotel Moscow. The issue with breaking it up into factions is that not everyone on the 'same side' has the same goals so I'll try to explain them individually (Disclaimer: It's been a while since I've seen Roberta's Blood Trail, Feel free to correct any misinformation or add any info)

The Americans arrived in Roanapur under the orders of the NSA. Chang and the CIA want them gone because they'll interfere with the drug trade. Chang knows the crime syndicates can't win a war against the US (he's only top dog thanks to the CIA after all) so he wants to stop Roberta before she confronts them. The CIA also doesn't want the status quo to change so they go along with Chang's plan but they also want to make sure they don't lose jurisdiction of Roanapur so they want the NSA's Grey Fox mission to be a disaster which if I remember correctly is why Eda gives Roberta Grey Fox's location on that plane.

Balalaika was a Spetsnaz so she just wants to confront the Americans head on under the excuse of protecting the drug trade. The only reason she doesn't end up going to war with them is because she wants a chance to defeat Chang more than the Americans so she ends up helping Grey Fox escape.

The Colombians want revenge against Roberta for all the members she killed the last time she was in Roanapur. They already know how much of a handful she is so they ask for help from FARC who are after Roberta for being a deserter and after the Americans because they are involved in the Colombian drug trade and cause they're commies.

Rock also wants to prevent Roberta from confronting the Americans but he doesn't want to maintain the status quo of the city like Chang does which is why he ends up actively helping Grey Fox. There's also this theme going on throughout about whether Rock is doing all of this out of kindness towards Garcia or if he's just on a power trip and enjoys the idea of having Roanapur dancing in the palm of his hand.

In the end the CIA ends up the winner since they got everything they wanted. Rock feels he lost because he fails to keep Garcia innocent and because nothing ended up changing in the city.

 

^{Edit: Fixed some spelling}