How do I incorporate an approval aspect into the form after is has been submitted?

Is there a specific app that would work best for email notification?  Also, once someone submits the form, how would that form be transmitted?

When a form is submitted, you will receive the Email Notification to your email address.

Setting up Email Notifications

You can also send an email to your users via Autoresponder Email. 

Setting up an Autoresponder Email

Submissions are stored on our servers, and you can view them on your JotForm account. 

How to View Form Submissions