A Valyrian sphinx is a mythical creature, with the body of an animal (most likely a dragon) and the head of a human. The Valyrians made statues of Valyrian sphinxes, some very large. Valyrian sphinxes are now relics, left over from Old Valyria, prior to the Doom. They are still seen in Essos and Westeros. They are still used as decoration and may be a sign of wealth and influence.
Whether they were made using advanced Valyrian techniques, such as spells and magic, or by more ordinary techniques is unknown.
A Game of Thrones
A Feast for Crows
A Dance with Dragons
While travelling, 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. Why the Dothraki did not take the queen sphinx is not mentioned. There is a smooth stone plinth where the king once stood, grown over with moss and flowering vines. Tyrion Lannister considers the queen sphinx a pleasant omen and her missing king being dragged back to Vaes Dothrak an omen too, but not as hopeful.