From A Wiki of Ice and Fire
Water Dancers are practitioners of a unique form of sword fighting uses by Braavosi Bravos called Water Dancing. The style is a refined form of fencing in which the practitioner stands sideways and wields a slender blade. Pugnacious bravos fill the city, frequently dueling to display their skill. While in King's Landing, Arya Stark trains as a Water Dancer under Syrio Forel, a celebrated Braavosi swordsman. It emphasizes quick graceful movements, stealth, balance, the sword being part of the arm, and seeing with all the senses.
Arya is on the steps of the Tower of the Hand when her father finds her standing on one foot with her feet scuffed. He asks what she is doing and she tells him that Syrio says that a Water Dancer can stand on one toe for hours. Eddard expresses his concern about her falling down the steps and Arya states a Water Dancer never falls.