I have installed this module http://www.nextbits.eu/check-delivery-availability-magento-extension.html and it is working in admin but not showing in front end. I have moved layout and templete files to base but still not working.
Have you tried the module with just the base theme on to see if it works?
If it does, then its more than likely a theme issue. If it doesn't work then its likely a setup error and you should probably contact the author of the module for advice.
Just to add to this, if you are deploying using symlinks (using modman for example) then you will need to make sure that you allow symlinks in the admin area.
If you are just putting the files directly in the design folder this will not be an issue.