The park's dirty little secret
An off trail hike in White Canyon inside Natural Bridges State Park
Do not do this hike if rain is predicted in the area. Also, minor bushwhacking will be encountered.
The area is rich in Anasazi ruins, please respect these sites and do not touch or otherwise approach them physically. Please admire them from a distance so they will be there with the same quality for the next generation.
Head down the Sipapu Bridge Trail, which is described by Rob already. After admiring the bridge, head upcanyon in White Canyon. This generally consists of walking along the streambed with occasional trails on the sandy banks on either side. Some bushwhacking will be encountered.
Within a mile you will see about a 5 structure dwelling in an undercut on the north side of the canyon. Admire them and continue hiking upcanyon. In another 1.5 miles you will reach Sphinx rock, distinctive in the way it is elevated in the middle of the canyon in a meander and vertically striped resembling the blue/gold stripes on Egyptian artifacts. To the north side is a massive Anasazi structure, including multiple dwellings and intact kivas. Respect the signs and chains the park service has put up and do not walk or touch these exquisitely preserved ruins.
If you are so inclined, continue hiking up White Canyon and possibly up Burch as well to find more artifacts and ruins of the Anasazi. You may find Kelsey's "Non-technical Canyon Hiking Guide to the Colorado Plateau" helpful if you do this, however I ran out of time. Statistics at the top are only to Sphinx Rock ruins and back.
Nothing you want to drink, bring plenty.
Not allowed in the canyon, though there is a campground area in the park, it was full upon my visit.
Check out the Official Route and Triplogs.