Along with being the name of the actual sons of Fëanor, the term is used in The Silmarillion and its associated material to describe a political entity, the elves that were led by the sons of Fëanor.
Someone trying to puzzle out the exact demographics, or political or economic structure of Tolkien's elves might spend many hours trying to piece together a picture based on sparse information. In the Silmarillion, it says that the sons of Fëanor dwelt in East Beleriand with "many people", but whether that many was several thousands or several million would be anyone's guess. Along with that, the question of why certain elves were bound to certain Elvish lords is not addressed. Were "The Sons of Fëanor" an extended family of sorts? Were they tied by some sort of system of feudal oaths? Did they share a common dialect or customs? These questions are not addressed, and are perhaps not too meaningful; and would deflate the myth through details.
Even in a mythic book, there is some political structure present in the Silmarillion, and the Sons of Fëanor, however they are constituted, play an important, yet paradoxic political role. The Sons of Fëanor, through their oath (and the text does state that only the seven actual sons of Fëanor, not their extended kin, took that oath), were the instigators and leaders of the coalition that fought Morgoth during the First Age. The paradoxical part of this is that although they were in some ways the core constituency of the alliance, they were also extremely antagonistic to almost all the other members, and they didn't contribute any notable military successes to the wars. This is an example of Tolkien representing something mythically that could also be a common political reality. According to the Silmarillion,
Caranthir was haughty and scarce concealed his scorn for the unloveliness of the Naugrim, and his people followed their lord.
although this is put in the mythological terms of an antipathy towards a different race, it is a very believable political situation, to have one ally distrust another ally because of relatively minor differences in customs. Along with this, I think Tolkien subtly paints the Sons of Fëanor as what we would now term
chickenhawks. They are the ones who take the oath to hunt down Morgoth to the exclusion of all else. Yet in the book almost every single valiant (or even useful) deed is done by their cousins, (and rivals) amongst the
Noldor, or by the
Sindar, or by humans, or dwarves. The one large military initiative that the Sons of Fëanor do put together against Morgoth turns into the disastrous fifth battle, the
Nirnaeth Arnoediad. After this battle, they "wandered as leaves before the wind". I think Tolkien was making some sort of comment that the same faction that would engage eagerly in civil war was also unable to actually raise any type of effective martial force against their real foe. Whether this is totally due to the personalities of the seven Sons of Fëanor, or was a trait of the culture or habits of the group around them, is unclear from the text.
I think the portrayal of the Sons of Fëanor shows the amount of irony and subtlety in matters of politics and morality that can be gained from the Silmarillion, if one reads closely enough.