The notification module uses the order hook system and other events to respond changes in an order by sending a notice of the change to the customer. The various settings pages let you configure the e-mail notifications for these different events. The only general setting at the moment lets you specify a store help URL for use in e-mail notifications so customers can come learn how to navigate and use your website as logged in users.
Invoices use templates that are either in files (in ubercart/uc_order/templates) or configured through the settings form. These templates allow the use of tokens which are simple pieces of text you put in the template that get replaced with actual values when the notification is generated. For example, including [order-id] in your template will show the actual order ID in the e-mail to the customer.
Ubercart comes with a default invoice template, but you should read this page if you're interested in creating and customizing your own template.
When a customer completes checkout, you can enable an e-mail notification sending them their order details and links to their account pages. Furthermore, you can send as many admin notification e-mails as you want by specifying multiple e-mail addresses in the admin settings text box. Ubercart provides a default customer template, customer.itpl.php, and a very basic admin template, admin.itpl.php.
When an order is updated and the checkbox to send an e-mail notification is checked, Ubercart will send an update notification with the current order status and the order comments to the user. Currently this is a very minimal form and does not use a full fledged invoice template file.
If you are using the uc_roles module to sell site access and membership roles, there will be another section pertaining specifically to notification messages sent by this module. These include e-mails that alert the customer when they are granted a role, when a role they purchased expires, when they renew a role by purchasing it again, and when a role they purchased is about to expire so they can come renew it.