A purr was a guttural vocalization made by various types of animals, often used by humanoids as an onomatopoeia. It was often done as an indication of pleasure or happiness. (TAS: "The Terratin Incident"; TNG: "Schisms")
In 2268, Nyota Uhura was delighted when the tribble she held began to purr. Cyrano Jones assured her that it meant the tribble liked her. (TOS: "The Trouble with Tribbles")
Cats were well-known for their purring, particularly when petted. In his poem Ode to Spot, Data referred to this sound as "a singular development of cat communications". (TNG: "Schisms") Reginald Barclay's cat Neelix also purred. According to Deanna Troi, this trait probably influenced Barclay's decision to make a holographic re-creation of the actual Neelix purr. (VOY: "Life Line")
Caitians often purred when they spoke. (Star Trek: The Animated Series)
Guinan's imaginary friend, a Tarcassian razor beast, had "the softest purr you'd ever heard." (TNG: "Imaginary Friend")
In 3188, Cleveland Booker's cat Grudge was purring as he and Michael Burnham were talking. (DIS: "That Hope Is You, Part 1")
The term could also be used to describe the sound of a motor running. In 2259 of the alternate reality, Montgomery Scott assured Captain James T. Kirk that USS Enterprise's warp core was "purring like a kitten" (i.e. running well). (Star Trek Into Darkness)
Montgomery Scott and Miles O'Brien both compared starship engines to kittens purring; Scott in 2269 (USS Enterprise), and O'Brien in 2370 (USS Prometheus). (TAS: "The Terratin Incident"; DS9: "Second Sight")
Tom Paris had once heard that California class engines purred. (LD: "We'll Always Have Tom Paris")