I actually remember the day I learned the word nip, when I was little girl. My brother and I were playing with the puppy and he grabbed my hand with his mouth. "Mommy! The puppy is biting me!" My mother said, "No, he's not biting you. It's just a little nip."
A "nip" is grabbing with the teeth but not breaking the skin and not intending to cause injury, just like what puppies and kittens do when they play.