Scabbers is Ron Weasley's pet rat. He is an old, rather useless rat and Ron pretends to hate him, but he is secretly very attached to the rat. Scabbers used to belong to Ron's older brother, Percy. Although the first years' shopping list only mentions cats, owls and toads as allowed pets, there does not seem to be any problem with Ron bringing him along to Hogwarts.
From the Story
"Scabbers is Ron Weasley's pet rat, who once belonged to Ron's older brother, Percy. Ron claims he's useless because he sleeps all the time, though he did wake up to bite Gregory Goyle in Ron's defence on the Hogwarts Express."
Discovered in Book 1, Chapter 6, The Journey from Platform Nine and Three-Quarters
"Ron claims Scabbers is useless because he sleeps all the time, though he did wake up to bite Gregory Goyle in Ron’s defence on the Hogwarts Express."
Discovered in Book 3, Chapter 4, The Leaky Cauldron (chapter)
"Scabbers is Ron's pet rat, a hand-me-down from his brother Percy. Ever since the Weasley family's trip to Egypt, Scabbers has been looking quite off-color. In the Magical Menagerie, Crookshanks attempts to attack the rat, and Scabbers promptly dashes out of the shop down Diagon Alley, taking refuge under a wastepaper bin outside Quality Quidditch Supplies."
Discovered in Book 3, Chapter 16, Professor Trelawney's Prediction
"Scabbers is found inside Hagrid's milk jug. He scrabbles to remain inside the jug as Hermione tips it upside down, but lands on the table. He struggles when Ron picks him up. Scabbers looks dreadful; he is thinner than ever, large tufts of his hair are missing, and he is covered in bald patches. He writhes in Ron's hands as though desperate to free himself. On the way up to the castle Scabbers goes beserk; he flails around and tries to bite Ron, squealing wildly."
Discovered in Book 3, Chapter 17, Cat, Rat and Dog
"Scabbers becomes more and more agitated as the trio make their way back towards the castle. He desperately tries to escape Ron's grip; wriggling madly, he bites Ron, and is clearly terrified of something. When Crookshanks appears, Scabbers slips through Ron's fingers and scampers away from the cat. Scabbers is caught by Ron, and is placed back in Ron's pocket. When Ron is attacked by the large, black dog, Scabbers is taken with him into the Whomping Willow.
Scabbers thrashes around desperately when Ron takes him from his pocket to show Professor Lupin. Lupin and Sirius Black tell Harry, Ron and Hermione that Scabbers is actually a wizard and an Animagus, and that his name is Peter Pettigrew."
Discovered in Book 3, Chapter 18, Moony, Wormtail, Padfoot and Prongs
"Scabbers desperately tries to escape from Ron, scratching at his face and neck in the attempt. Squealing like a pig, he struggles in Ron's hands, but can't get free."
Discovered in Book 3, Chapter 19, The Servant of Lord Voldemort
"Scabbers struggles in Ron's hands, still trying to get away. When Ron finally hands him over to Lupin, Scabbers begins to squeak continually. He twists and turns desperately, eyes bulging in his head, as he tries to escape. Scabbers is held tightly in Lupin's hand as he and Black prepare to cast the spell that will reveal whether he is an Animagus or not. There is a flash of blue light and Scabbers is momentarily frozen in mid-air, still twisting madly, before he drops to the floor. There is another flash of light before Scabbers quickly begins to change: a head rapidly shoots upwards, limbs sprout from his body, and the next moment the rat is replaced by a man.
Scabbers is revealed to be Pettigrew, the man Black was accused of murdering."