From A Wiki of Ice and Fire
A sphinx is a mythical creature, made up of a bit of this, and a bit of that: a human face, the body of a lion, the wings of a hawk. 
In our world, and most likely GRRM’s, some traditions believe that sphinxes take away the sins of devotees when they enter a temple and ward off evil in general. A sphinx or sphinxes are therefore often found in a strategic position, for example, near entrances or a temple gateway. Often they are found as female-male pairs. Here, too, the sphinx serves a protective function.
Sphinxes always speak in riddles in the tales. Tyrion mentions to Brown Ben Plumm that most of the stories heard about dragons are fodder for fools - one that he mentions is of dragons riddling with sphinxes.
A Game of Thrones
A Feast for Crows
|“||"The sphinx was the riddle, not the riddler".||”|
Once in Oldtown Samwell Tarly enters the gates of the Citadel which are flanked by a pair of towering green sphinxes. They have the bodies of lions, the wings of eagles, and the tails of serpents. One has a man’s face and the other a woman’s. 
A Dance with Dragons
While traveling Tyrion and Illyrio Mopatis come across a huge Valyrian sphinx crouched beside the road. It has a dragon's body and a woman’s face. However her king is missing, dragged off by the Dothraki back to Vaes Dothrak on wooden wheels. There is a smooth stone plinth where he once stood, grown over with moss and flowering vines. Tyrion considers the queen sphinx a pleasant omen and her missing king an omen too, but not as hopeful.
The Long Bridge, which joins the two halves of Volantis across the mouth of the Rhoyne, has a gateway which is an arch of black stone carved with sphinxes, manticores, dragons and other strange beasts.
|“||"The sphinx is the riddle, not the riddler," he blurted. "Do you know what that means?"||”|