Old English docga, a late, rare word used of a powerful breed of canine. It forced out Old English hund (the general Germanic and Indo-European word; see canine) by 16c. and subsequently was picked up in many continental languages (e.g. French dogue (16c.), Danish dogge), but the origin remains one of the great mysteries of English etymology.
Many expressions -- a dog's life (c.1600), go to the dogs (1610s), etc. -- reflect earlier hard use of the animals as hunting accessories, not pampered pets. In ancient times, "the dog" was the worst throw in dice (attested in Greek, Latin, and Sanskrit, where the word for "the lucky player" was literally "the dog-killer"), which plausibly explains the Greek word for "danger," kindynas, which appears to be "play the dog."
Slang meaning "ugly woman" is from 1930s; that of "sexually aggressive man" is from 1950s. Adjectival phrase dog-eat-dog attested by 1850s. Dog tag is from 1918. To dog-ear a book is from 1650s; dog-eared in extended sense of "worn, unkempt" is from 1894.
Phrase put on the dog "get dressed up" (1934) may look back to the stiff stand-up shirt collars that in the 1890s were the height of male fashion (and were known as dog-collars at least from 1883), with reference to collars worn by dogs. The common Spanish word for "dog," perro, also is a mystery word of unknown origin, perhaps from Iberian. A group of Slavic "dog" words (Old Church Slavonic pisu, Polish pies, Serbo-Croatian pas) likewise are of unknown origin.