ArcGIS shapefiles and Personal Geodatabases (mdb) files are readily compatible with ArcGIS.
File geodatabases are less compatible, although it is possible to open them if you install the driver using the OSGEO installer.
In the OSGEO Installer, install the “dal-filegdb: OGR FileGDB Driver” library. Restart QGIS and add in a vector file as normal. At the “Add Vector Layer” Screen, tick the option for “Directory” and stipulate the source as the geodatabase you want to open, as shown here.
When you click “Open” you will be presented with a list of all the datasets contained within the geodatabase. You can add these in by clicking on the one you want and clicking “add” or “open”.
NOTE: This requires the OSGEO Install of version 2.2 of QGIS – Version 2.0 contains a bug.