The Canvas Inbox displays all Canvas Conversations - the internal messaging system in Canvas.

Benefits of using the Canvas Inbox and Conversations are below!

Clean: Not a new email address
Easy to use: Connected to your main email address; can receive copies of your Canvas Conversations at the email or any Notification pathway and answer right from there. Will appear to students that the communication was all contained in Canvas
Purposeful: The only people who can contact you through a Canvas Conversation are those who are connected to you through a Course or a Group
Contacts loaded: Users attached to your Course are pre-populated in your address book

Here are some additional ideas about why and how using Conversations can make your life easier:

Conversations has a simple interface for composing and includes the ability to easily create an audio or video message
Archive instead of delete to always retain copies of Conversations with students

Need more information? Check out this Canvas Guide!