Cats purr most often when they are content, such as while being petted or resting somewhere cozy, but they also purr when frightened, in pain, or sick, which suggests purring also helps a cat soothe itself, much as people hum under stress (Britannica, “Why Do Cats Purr?”). So a purr is a useful clue, but not proof, that a cat is happy.
The sound comes from the larynx and continues as the cat breathes in and out. Domestic cats purr at roughly 25 to 30 vibrations per second. Scientists long credited rapid twitching of the laryngeal muscles, but a 2023 study found that cat voice boxes contain small pads of tissue in the vocal folds that can vibrate at purring frequencies with airflow alone (Herbst et al., Current Biology, 2023). Purring is shared by smaller cats such as cougars and bobcats, but not by roaring cats such as lions and tigers.
Source: Encyclopaedia Britannica, “Why Do Cats Purr?”.
Image: “Sleeping cat on her back” by Umberto Salvagnin, CC BY 2.0, via Wikimedia Commons.