Also spelled T'ien-kow, T'ien-k'uan and Tian-gou. The Japanese corruption of the T'ien-kou is Romanised as 'tengu'. This node is about the Chinese version, the heavenly dog (also translated as the celestial dog).
In Chinese mythology, T'ien-kou lived in the sky, acting as the guard dog for Erh Lang, a nephew of the Jade Emperor (AKA Yu-huang, the Great Emperor of Heaven). Erh Lang sics T'ien-kou on evil spirits that try to approach the Emperor's place. T'ien-kou also caused solar and lunar eclipses by swallowing the sun or moon; when this happened the mortals down on Earth would beat gongs and drums to try to frighten the dog away.
T'ien-kou is associated with meteors, Sirius, and storms. These appearances, along with eclipses, tend to be manifestations of T'ien-kou's darker (yin) side. T'ien-kou is somehow identified with the mountain demons, although I don't understand how this works -- when speaking of the mountain demons, T'ien-kou suddenly turns plural and evil*. Owners could also protect a cherished dog by placing an image of their dog on Erh Lang's alter; this is probably some sort of evocation of T'ien-kou.
When the legend of T'ien-kou was introduced to Japan the mountain demon turned into a bird, and these birds were structurally commingled with the Japanese yamabushi (mountain priests). At this point they became more mischievous than evil, and often would help humans. You should see the tengu node for for information on this.
T'ien-kou is also one of the many names given to the Chinese Foo Dog, a breed of dog somewhat related (and possibly ancestral) to the Chow Chow.
*I assume that there is a connection through meteors. Meteors, with there fiery tails, were thought to be dog-like, and connected with T'ien-kou, but also demon like, and connected with the mountain demons. Why these two creatures share the same name, I do not know. The Japanese tengu comes from the mountain demon, not the guard dog.