The most popular cave, and the only cave in the Park visitors are allowed to legally access without special permission is Cave of the Domes. This cave is a difficult 3.5 mile hike down the Grandview Trail and across Horseshoe Mesa to a small entrance about .3 miles off the trail (you wont find this without directions). Two other well know caves are Tapeats Cave, and Thunder River Cave which are located below the North Rim of the Canyon off a rugged backcountry trail.
There are several reasons the Park Service wont let visitors access these caves, but the two main reasons are the archeologically and biologically sensitive items found in the caves. With a lack of resources to explore and monitor caves systems within the canyon this probably wont change any time soon. Typically the only people who get permission are members of Caving Organizations who will agree to do mapping of unexplored areas for the Park in return for being granted permission. Wet Caves, like Tapeats and Thunder River, are not very difficult to receive permission to explore if you are an experienced Caver. The Grand Canyons Park Hydrologist is currently the person who makes the call on access to Grand Canyons subterranean world.
Contact John Rihs, the Park Hydrologist for additional information (928) 638-7905.
Or check out www.nps.gov/grca/naturescience/cave