Established Factions and Groups

Organised governments
The most influential and powerful form of faction, an existing organisation that governs its own peoples and has its own military or acting-militia force. These groups more easily obtain wealth and territory, while being able to influence or intimidate others with their size. They have established hierarchies, laws, and culture, and command over their territories. They have their own flags, banners, or insignia, which are worn to brand their militaries or territories. They can establish trade routes, travelling caravans, or defensive checkpoints which will not be easily overtaken by a lesser group. They qualify as factions with 'great power' and cannot easily be extinguished. 

• The New California Republic
• The Brotherhood of Steel
• The Legion
• The Institute
• The Enclave

Not exactly second to the organised governments, but follows closely behind. A cooperative affiliation marks the organisations which cannot be classified as an outright government, due to a great number of things. They may lack territory, culture, law, numbers, or what have you, but still are influential enough to qualify as a faction. They may obtain assets, territories, or followers together, however they are not considered uniform or a fully functioning society. They may have militia or small militaries, or trade routes, however they do not have the resources to defend a large segment of territory.

• The Great Khans
• The Minute Men
• The Railroad
• The Dead Horses
• The White Legs
• The Southern Utes - last remaining member, Kaw
• Independent cities/settlements/stationary tribes

Territories
Claimable territories are the physical boards on the forum that represent an area of the wastelands. Each territory may be claimed, although they may be canonically controlled by a faction and therefore very difficult to assume power over; a large, organised government which historically controls that area will be backed up by the lore masters, who will make it difficult if not impossible to lay claim to areas that are canonically claimed (ie the Brotherhood's citadel).
How do you claim a territory? Roleplay for your life.