I would like to send an email to the signed user with the data that was added by the user to the Firebase database. I am using AngularJS UI. Is there are Firebase module which will trigger email with the data being added to the signed user.
You should look into Cloud Functions for Firebase. This will let you write database triggers that run on Google servers that can act whenever data in your database changes. There is plenty of sample code the illustrates some of the many things you can do, including one specific example that shows how to send email in response to a change in the database.