Crookshanks is the cat bought by Hermione Granger at the beginning of the book Harry Potter and the Prisoner of Azkaban. He is known to be very clever and had the ability to communicate with Sirius Black when he was in he animagus form.
 Prisoner of Azkaban
Before the start of the school year in 1993, Hermione Granger purchased Crookshanks from Magical Menagerie in Diagon Alley. She had been given money to buy herself a present and discovered Crookshanks when he leapt onto Ron Weasley's head, who was having his pet rat Scabbers looked at. When buying the cat, Hermione was told that he had been there for ages and no-one wanted him. Ron was less than pleased that Hermione had taken pity on him.
Being part Kneazle (see below in Canon), Crookshanks sensed that Scabbers, Ron's pet rat was not to be trusted and took an immediate dislike to the rat. The pair had several fights throughout the year and eventually Crookshanks was falsely accused by Ron Weasley of killing and eating Scabbers, who himself has a dubious reputation. Because of this accusation, Ron and Hermione had one of the worst fights they had ever been in, and Ron was constantly accusing and reminding Hermione of the fact that Crookshanks supposedly ate Scabbers. Ron later got over his hatred of Crookshanks when he realized that Scabbers was indeed alive.
Scabbers the rat was found to be the secret Animagus form of Peter Pettigrew, a follower of Lord Voldemort, who revealed the secret of where Harry's parents were hidden. Crookshanks' name was cleared when Rubeus Hagrid gave Scabbers back to Ron in his hut just before the Minister of Magic, Cornelius Fudge and an executioner arrived to behead the hippogriff, Buckbeak.
Crookshanks also consorted with a mangy-looking dog who turned out to be the Animagus Sirius Black, Harry's god-father who had been falsely accused of murdering Peter Pettigrew and numerous muggles. For this crime Sirius had been sentenced to the Wizard prison, Azkaban. He met with Sirius several times on the lawns of Hogwarts castle and even stole the passwords to the Gryffindor common-room for him.
When Harry and Hermione are stuck not knowing how to rescue Ron from the dog, not yet knowing it was Sirius, Hermione calls out for help and Crookshanks shows them the way to get past the formidable Whomping Willow. He darted under the attacking branches and pressed a knot on the trunk, causing the tree to freeze.
In the books following Prisoner of Azkaban, Crookshanks has a minor role and is only seen in minor references.
 Outward appearance
Crookshanks is described as a lion-like, bow-legged ginger cat with a squashed-together face. He also has a large bottlebrush tail.
 Personality and Traits
Tends to bother Ronald Weasley's pet, Scabbers. He is unusually intelligent, understanding what people want him to do, and is very loyal to Hermione.
 Additional Canon
Crookshanks was inspired by a real cat that J. K. Rowling had seen.
When I was working in London in the late 1980s I used to eat my lunch in a nearby square on sunny days and a large, fluffy ginger cat that looked as though it had run face-first into a wall used to prowl around the sunbathers there; I assume it lived in a nearby house. I didn't ever get close enough to give myself an asthma attack, but I became distantly fond of this cat, which prowled among the humans around it looking disdainful and refusing to be stroked. When I decided to give Hermione an unusually intelligent cat I gave him the appearance of this haughty animal, with the slightly unfair addition of bandy legs.
Also J. K. Rowling said in the same section on her website that Crookshanks is a half-breed: half cat and half Kneazle.
Crookshanks, as anybody who has read Fantastic Beasts and Where to Find Them will have guessed, is half Kneazle. And if you don't know what a Kneazle is, you need to hurry up and buy Fantastic Beasts and Where to Find Them (all royalties go to help some of the poorest children in the world).
Kneazles have the ability to recognize suspicious characters, which is what led Crookshanks to attack Scabbers (as he was an Animagus) and to befriend the Animagus, Sirius Black.