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 an easy prey until few domestic specimens were left, although some still survive as witnessed in the Commonwealth.
Rumor is that the last known living cat in America, Mr. Pussy, died in Carbon, Texas in 2208. Ruby the prostitute wanted to find him when he got lost, and asked the Initiate to help. They discovered the cat's corpse in the Carbon Crater and brought it back to Ruby.
If you use the perk Animal Friend on a cat and attack or get 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.
Fallout 4 is the first canon game in the series in which cats are actually seen alive. (Fallout Shelter being non-canon.)
Killed cats will usually have Cat meat in their inventory.
For additional information check out the House cat.