What do I have to do in order to be able to receive notifications? In the event of a database error in an application, what do I have do in order to be able to receive notifications?

Should I create an alert in SQL Server and use the RAISERROR command within the application?
I'd say that's a pretty way to go about it.  You can create a new user-defined error message using sys.sp_addmessage, which will in turn store it in sys.messages.  And then when you add your alert, you can specify to alert on a specific @message_id.  This can be seen through the definition of sp_add_alert.

Then when you have this alert in place and the error message is raised, you should get a notification provided that side is all set up correctly.