With most computer operating systems, there should be a function to assign a file extension to open with a particular application. This sort of problem is usually an Operating System level problem.
To correct this, as an example on my computer running Mac OS 10.7:
From the File menu -> Open With -> Other...
Browse to your Applications folder and click QGIS. At the bottom of the Window, click the "Always Open With" check box.
Click Open. At this point, all .qgs files should now open with QGIS.
Even if you don't have a Mac, the steps involved are similar with most operating systems. Usually the screen required to make the change is accessible by right clicking on the file.
If on a Mac and the above did not work, try running the two following terminal commands:
/System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/lsregister -u /Applications/QGIS.app
/System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/lsregister -f /Applications/QGIS.app
(The Terminal application can be found in the Utilities folder. The Utilities folder can be accessed from the Go Menu while in Finder.)
If you have QGIS located in a different spot other than the Applications folder, you will need to adjust the path accordingly. These two commands will unregister all filetype associations then force the Application to re-register them.
Once you have run the second command, you should be able to double click on .qgs files and have them open correctly.
Information courtesy of: http://krypted.com/mac-os-x/lsregister-associating-file-types-in-mac-os-x/