In the geospatial world a directory ending in .gdb is usually the Esri File Geodatabase FormatEsri File Geodatabase Format, which has an open APIhas an open API which many 3rd party programs can read -- gdal/ogr, fme, and GlobalMapperGlobalMapper to name the ones I'm familiar with. Gdal/ogr is open source and is included in many other applications like QGIS and SAGA.
Windows end users can most easily get the filegdb driver via the OSGeo4WOSGeo4W gdal-filegdb package.
For developers, the gdal/ogr FileGDB driverFileGDB driver is not included by default as the Esri SDK needs to be downloaded and installed separately (see build instructionsbuild instructions).
A file ending in .gdb is likely the Garmin GPS Database format used by MapSource and BasecampMapSource and Basecamp and the open source gpsbabel can read it, see Batch converting Garmin GDB to GPX?.