A hound is a type of dog with big floppy ears, mainly used for hunting. To hound someone is to relentlessly pursue or pester them. When Elvis sings, “You ain’t nothing but a hound dog,” he’s referring to both.
Hounds are dogs that have traditionally been used for hunting because they are excellent at tracking. Most hounds (like beagles and dachshunds) have drooping, floppy ears. Hounds can also be big and scary like the one Sherlock Holmes investigates in The Hound of the Baskervilles. If people hound, they hunt, too, but just in an annoying way. A man can hound a woman by asking for a date 10 times. A hounding person is persistent and probably annoying.