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 George R. R. Martin’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.
A Game of Thrones
A Feast for Crows
The sphinx was the riddle, not the riddler.
Once in Oldtown Sam 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 Lannister 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. Do you know what that means?
References and Notes