Note: Vaults aren't included because they don't represent post-war built societies.
Tier 1
New California Republic
Ceasar's Legion
Midwestern BoS (formerly?)
Tier 1 is made up of groups that have largely transcended mere survival, and are the closest equivalent to modern nations. They have populations in the hundreds of thousands or more, and they are capable of holding and maintaining extremely large territories. They have very complex governments with a well defined chain of command. They have their own standing armies and could defeat any civilization in the lower tiers in an all-out war. While they often possess some very advanced pre-war technology, they usually don't use it as extensively as others, partially because of the impracticality of using such technology on a large scale. They will usually be expansionist in nature, believing that they are meant to bring their form of civilization into the wasteland.
Tier 2
Brotherhood of Steel
Enclave
New Vegas
Boomers
Midwestern BoS
Pitt Raiders
Tier 2 is composed of societies that have a significant size and level of organization. Their economy and population are usually much larger than just that of a single settlement. Often they will be centered around a relatively large headquarters or city that is under their control. They usually don't control a lot of territory compared to Tier 1 but they may have bases and outposts in several different locations, or have influence over a large area. They also possess a large amount of pre-war technology, including weapons. While they would still lose to Tier 1 societies in an a war, they would inflict very heavy casualties on them; because of this they are often taken very seriously by others.
Tier 3
Megaton
Great Khans
Rivet City
Goodsprings
Novac
Jacobstown
Primm
Utobitha
Underworld
Nipton (formerly)
Arefu
Big Town
Canterbury Commons
Tier 3 is roughly the equivalent of a small city-state. They're generally a small town or community that has some way of supporting themselves, either through scavenging, hunting, trading, farming, or a combination of these. They usually aren't armed to the teeth, but they have enough weapons to protect themselves against raiders and wild animals. Unlike raiders they are also rarely hostile to outsiders without a reason. They are the lowest tier that represents a functioning society that can exist in the long term, although they may not always stay independent.
Tier 4
Fiends
Jackals
Vipers
Little Lamplight
Powder Gangers
other independent raiders
most tribals
Tier 4 is made up of groups that are the least stable. They may be only a small group of raiders who have no real plan for the future. They're economy is often based on raiding or saluaging at best, and they're often antagonistic towards neighboring societies. While tribal groups of hunter-gatherer/scavengers can remain independent until civilization reaches them, the inherently violent nature of raiders often forces more civilized societies to deal with them relatively quickly. In the long run these groups will either gain a more permanent settlement and way of life or disappear completely; crushed by the onslaught of civilization or devoured by the wasteland.