You could create a base activity (BaseActivity class) from which all your activities would have to be derived. Then inside BaseActivity.onCreate - you would increase some SharedPreferences counter, and inside BaseActivity.onDestroy you would decrease it. Now when that counter is equal to zero, you might assume your application is closed - but the process might still run in the background.
What I want is receiving something from the database
now I assume you already know how to "receive something from database". This could be background Service, where you could check SharedPreferences and do your processing.