If you want to add a flag to your Facebook profile, you could upload a picture of a flag to the photos section of your profile. However, that limits your flag display to one place -- your photo gallery. If you want a more versatile placement of flags, use a Facebook app. Apps are third-party programs that you can use to add elements such as flags to your profile.
Navigate to the Facebook App Directory, http://www.facebook.com/apps/directory.php, and type "flags" into the "Search Apps" text box.
Click on an application name. There are several applications that will add flags to your profile including "World Flags," "Flags!" and "Painted Flags."
Click "Go to App" and then click "Allow" to give the app access to your profile.
Follow the instructions to use the app. Each app is different but in general, a pop up window will appear as soon as you install the app.
If you want to use the app again to add more flags, click the app name in the left hand column of your Facebook profile.