Find the source of the leak. Sometimes the puncture or tear is large enough to spot quickly. On occasion, the hole is tiny and difficult to see.
Use your ears when you cannot find the hole. Fill up the air mattress with air, and listen for the hissing sound that results from air exiting through a small hole.
Mark the hole with a marker. Deflate the mattress completely.
Use a heavy-duty glue product on the air mattress hole if it is small. The best place to purchase a glue that will stand up to the stresses put on a mattress is a dive shop. Glues at hardware stores can also do the trick.
Apply glue generously around the hole. Make a circle with the glue of about an inch in diameter with the hole in the center.
Let the glue dry for at least 24 hours.
Use a water bed repair kit to repair large punctures and rips in an air mattress. These kits typically involve a patch and glue. They are built to hold water in a mattress so they probably can do the same with air. Kits can be purchased online or at a bedding store that sells water beds.
Test the fix. Fill up the air mattress and leave it out for 12 hours. Come back and check whether the air mattress has held air.