You've published your website and it can't talk to your database because it's not you any more, it's now running as the AppPoolIdentity. You could set it to run as you but that's far too much permission for production.
You could also create a user, if you have the authority, but it can get messy when multiple domains are involved.
Or you could grant permissions to the default app pool identity, if you knew how.
CREATE LOGIN [IIS APPPOOL\name_of_app_pool] FROM WINDOWS WITH DEFAULT_DATABASE=[name_of_relevant_db];
CREATE USER [IIS APPPOOL\name_of_app_pool] FOR LOGIN [IIS APPPOOL\name_of_app_pool];
At this point both login and user are visible in SQL Server Management Studio and you can interactively assign database permissions.