The majority of GPS navigators and FMS have a dataloadable navigation database. The actual database each uses is typically encrypted and compressed using proprietary algorithms. The source data for all of these databases normally start out in ARINC 424 format. Unfortunately, A424 is not available free of charge.
ARINC 424 is an international standard file format for aircraft navigation data maintained by Airlines Electronic Engineering Committee and published by Aeronautical Radio, Inc.. The ARINC 424 specifications are not a database, but a "standard for the preparation and transmission of data for assembly of airborne navigation system data bases".
The databases are compiled by companies such as Jeppesen. Jeppesen collects the data from the controlling agencies around the world, checks it, and merges it into a world-wide database that they then sell (typically via subscription).
Some information for US airspace is available directly from the FAA. It is not in a format that can be directly loaded onto aircraft, but may be of use offline.