After a dive, the diver's blood is saturated with dissolved nitrogen, resulting in the formation of microscopic nitrogen bubbles throughout the body. As long as they remain microscopic, these bubbles are harmless.
The dive tables assume the diver will be returning to and remaining at the surface, under normal atmospheric pressure. However, flying exposes a person to reduced pressures, which bring additional nitrogen out of the blood and allow the nitrogen bubbles to grow in size. If they become too numerous or large, they can begin to gather together, resulting in a host of problems.
While there are no legal requirements concerning flying after diving, the FAA published recommended waiting periods between diving and flying. For a normal SCUBA dive, at least 12 hours is recommended between the dive and the flight. If the flight is to be above 8,000 feet, or if the dive was a decompression dive, then a waiting period of at least 24 hours is recommended.