Quite simply, a "running dog" is a dog bred for running, e.g. a greyhound. In fact, the name "greyhound" may have come from the Saxon word "Greu", which means running dog. Figuratively, however, "running dog" has a derogatory usage, which means lackey or sycophant. This insult is closely associated with the Communist leaders of old, especially Mao Zedong, who would refer to various USA-friendly states as the running dogs of imperialism.
A running dog, be it a person or state, was a servile entity which did the bidding of greater powers, in order to stay in their good books. The analogy is pretty clear: in exchange for food, protection and other rewards, a greyhound will mindlessly run great distances at great speeds in pursuit of some fabricated lure.
The capitalist side, on the other hand, preferred to use the term puppet to refer to states in a servile relationship with one of the Communist powers. Nowadays, "running dog", "imperialist running dog", or "capitalist running dog" are generally used as amusingly long-winded insults with a retro feel harking back to the days of the Cold War.