Cats are a species of small predatory carnivorous animals that were valued by humans before the Great War for their companionship and ability to hunt vermin, such as rodents, snakes, scorpions, and other household pests.
After the Great War, many cats in what used to be the United States were hunted for food because they were easy prey. Although not as common as before the Great War, domestic cats still survive as seen in the Commonwealth.
The following is based on Van Buren and has not been confirmed by canon sources.
According to the Nursery design document, various cats were to make an appearance there. There would have been friendly domestic cats kept by humans, as well as wildcats, bobcats and panthers that were hostile to the player.
Using the perk Animal Friend on a cat and attacking or getting caught stealing from a settlement, all settlers will attack the cat as though it is a threat, though it is not combat-enabled.
With the Wasteland Workshop add-on, a cat can be captured in any settlement via a cat cage. Once released, it will add 10 bonus happiness points to the settlement (as much as some tier 1-2 stores). A beta wave emitter is not necessary to tame them. Also unlike all other captures, cats captured this way are protected.
Fallout 4 is the first canon game in the series in which cats are seen alive. (Fallout Shelter being non-canon.)
Killed cats will usually have cat meat in their inventory.
Cats are not hostile to the player character in any way.
The following is based on unverified behind the scenes information and has not been confirmed by canon sources.
Cats are one of the few creatures in any Fallout game to be completely unaffected by radiation, at least externally. Some cats were possibly thrown into FEV vats by the Master in the Mariposa Military Base and by the Vault-Tec scientists in Vault 87, together with humans, dogs and other animals. The misshapen centaurs are the result of this mix.
It was often thought in the fandom prior to Fallout 4 that cats were rendered extinct by overhunting by wastelanders. Whilst Stacy in Fallout 2 does say there are no cats around here anymore, she didn't specify whether she was talking about the USA as a whole or simply her home in The Den. Several NPCs from richer towns within the same game also talk about cats, implying they still existed.
End of information based on unverified behind the scenes information.