Say you’re writing a story in which your main character busies herself with petting a saber-toothed Martian cat. You want to indicate that only one vermillion-furred beast with extended maxillary canine teeth and a habit of hissing poison exists in the scene. Therefore, as your character, Zelda, reaches out to pet the cat, you’d write, “The Martian cat, which Zelda reached out to pet, suddenly bared its saber teeth and hissed poison at her.”
In this sentence, by using the word which and by setting the phrase which Zelda had been petting off with commas, you can indicate to the reader that only one extraterrestrial feline exists in this scene. The cat is skulking around the spaceship, and oh, by the way, Zelda pets it before the unusual feline turns on her.
On the other hand, suppose that Martian cats surround Zelda but that only the one that she had been petting has turned on her. How would you indicate this? You’d write, “The Martian cat that Zelda had been petting suddenly bared its saber teeth and hissed poison at her.”
By not setting the phrase off with which and its attendant commas, but by instead using the proper “that”, you easily indicate the cat that out of all the felines in the scene has turned on Zelda – the one that she so foolishly petted. Let’s hope that Martian cats prefer tuna fish!