The domestic cat or "house cat," known to biologists as Felis catus, is a sub species of Felinae and is itself a sub-class of the larger class Felidae. The house cat's closest relatives include the wildcat and the Chinese Mountain cat. Felis catus, based on a ceremonial burial in Cyprus, seems to have been a human companion for at least 9,000 years, or since the Neolithic era (think Stonehenge and earlier, thousand of years before the pyramids). Technically, wild cats or Felis silvestris are the ancestral species of the smaller Felis catus; the primary distinction being that Felis catus has been selectively adapted by humans.
Felis silvestris, the Wildcat is a small cat, albeit one that is larger than all but the very largest of domestic cats. It is native to Europe, the western part of Asia, and Africa, and like its domestic relative, it hunts mammals, birds, and other creatures of a similar or smaller size. Wildcats are currently divided into five extant subspecies: the European wildcat, the Near Eastern wildcat, the Southern African wildcat, the Central Asian wildcat and the Chinese desert cat. Recent DNA research has demonstrated that all of the domestic breeds we think of as house cats or Felis catus are descended from one of five female of the Wildcat species in the Near East known as Felis silvestris lybica, first domesticated about 10,000 years ago. The DNA of all house cats and fancy breeds of cats are descended from this Near Eastern Wildcat species, according to DNA studies by Carlos A. Driscoll published in the journal Science in 2007.
About the time of that Neolithic burial in Cyprus, farmers were growing semi-domesticated grains, especially wheat, rye and barley. Growing grain inevitably leads to storing grain, a practice that attracts rodents. Rodents attract cats, and so it seems quite likely that the conditions were perfect for a human-feline partnership to be born. One of the things that's particularly compelling about that kitten, apparently buried in a grave with its human, is that Cyprus was settled by Neolithic farmers who crossed over to the island from Turkey, and would have brought all their domesticated animals with them, including cats (there is no evidence at all of any native wildcats in Cyprus).
Image credit: Digital Medievalist; from the collections of the Metropolitan Museum of Art.
C. 332–30 B.C. Egypt's Ptolemaic period