Fluffy was very dangerous, very large three-headed dog owned for a short time by Rubeus Hagrid. About a year before Harry’s first year at Hogwarts, Hagrid purchased Fluffy from a Greek man in one of the local pubs (probably the Hog’s Head). (PS11)
Fluffy’s weakness was music, which put the creature to sleep. After lending him to Dumbledore to help protect the Philosopher’s Stone (PS11), Hagrid released Fluffy into the Forbidden Forest (BP). Following the Battle of Hogwarts, Fluffy was sent back to Greece (JKR:Tw).
Fluffy seems like a rather common and unimaginative name (more like the name of a cat than a dog). In Greek mythology, Hades called his three-headed dog Kerberos (Latinized to Cerberus), which is believed to mean "spotted". So the original inspiration for Fluffy had an equally uninspired name. (source: Wikipedia)
J.K. Rowling said on Twitter that Fluffy was expatriated back to Greece instead of sent to the Forbidden Forest:
.@EmyBemy2 He was repatriated to Greece. Dumbledore liked to put Hagrid's more foolish acquisitions back where they belong - not the forest.
— J.K. Rowling (@jk_rowling) February 6, 2015
In Greek mythology, Cerberus is a three-headed dog who guards the entrance to the Underworld (Wikipedia). -BB
The name "Fluffy" is humorous because it is usually thought of as fitting for tame, lap-sized pets. And this Fluffy is certainly neither of those things. It highlights Hagrid's love for and tenderness towards large, dangerous beasts, and contrasts with Hagrid's own gentle nature despite his size and intimidating appearance. Additionally, this mismatch of name and characteristics complements the same in Hagrid's other dog, Fang, who is gentle, friendly, and a bit timid. -BB
Hagrid purchased Fluffy from "a Greek chappie" because the three-headed dog is found in Greek mythology. This was apparently lost on Steve Kloves, the scriptwriter forPS/f, since he changed it to "Irish chappie" without catching the connection. To be fair, he did say that he had originally made it "Greek chappie" and that he didn't know how it got changed, but it does say "Irish" in the script itself, so it wasn't just that it was changed by Robbie Coltrane on the set and no one caught it.