A faction can best be thought of as a sub-civilization ... it is a population unit within a Civilization that has goals that vary from the mother civilization. A specific faction will tend to have levels of power that vary greatly from world to world ... it may have its power concentrated in one colony or world, or it may have influence in several worlds. It has resources of its own, but depending on how tightly a civilization controls its population, it can be difficult to amass enough resources to make independence from its mother civilization within the realm of possibility.
Factions will tend to have an easier time where government controls are weaker or the government is seen as failing the people, as the population will be more responsive to a locally strong faction if it tells them what they want to hear if the population feels they are not getting all they need, or they do not like what their government is doing. A xenophobic faction may grow stronger if the population mistrusts government dealings with an alien civilization, for instance; a warmongering government, on the other hand, will find pacifistic factions growing stronger especially when a war becomes costly or the people become shocked at abuses committed upon beaten enemies. Factions with sufficient resources can infiltrate local governments or even the national government and military, and then they become truly dangerous with the capability of launching a significant civil war (one or more factions declaring war on the mother civilization).
Government controls will tend to be weaker in fledgeling worlds and worlds with strong civil police forces; this will allow factions to become stronger in such areas.
Factions do not always have independence as a goal; they may wish to better a civilization and during a time of war, loyal factions can help boost the war effort by contributing the resources they are amassing. On the other hand, however, some factions may take advantage of war to strike out on their own with their motherland's national government focusing their military efforts on the alien battlefront.