Bryce Canyon is not, technically, a canyon -- a canyon is carved from a single river, and, while the geology of Bryce does involve erosion by water, a single river was not responsible.
Instead, the land was uplifted and then water from several streams and lakes carved the landscape 63-40 million years ago. The rounded pillars, or hoodoos, have been chemically weathered by a weak solution of carbonic acid which forms when rain combines with carbon dioxide. The carbonic acid gradually removed the softer limestone and rounded the rock's sharp corners, leaving the hoodoos standing like sentinels.
Bryce Canyon is also a part of the Grand Staircase, a series of "steps" that increase in elevation from the lowest at the Grand Canyon (around 7000 feet), through the middle elevation at Zion, to the highest here at Bryce (around 8000-9000 feet).