eClass allows you to choose whether to be notified by email and/or pop-up messages for a number of system events such as direct messages, new forum posts to which you are subscribed, and updates for quizzes/assignments. By default, these notification settings are enabled.

Before you change them, be aware that many instructors use the Course Announcements forum, to which you are automatically subscribed, for communicating course information.
If you decide to disable email notification for subscribed forums, you will need to check the Announcements forum regularly.
If you wish to receive email notifications from specific forums, consult with the following article on managing your forum subscriptions: Receiving Notifications from Forums (Subscription)

Setting Message Preferences:

To change these settings, click Preferences from your user dropdown menu on the upper-right. 

To change your notification settings for direct messages between users, click Message Preferences to slide out the message preference settings from the right.

You can choose to allow messages only from contacts (users who you have specifically added in the messaging interface) or from contacts and anyone in any curse where you are enrolled.
You can also choose whether or not to receive emails notifying you of new personal messages.

Viewing Messages

The messaging drawer is accessed via the message icon in the top nav bar. There will be a number shown, corresponding to the number of notifications or unread messages you have.

Clicking on the messaging icon will display an overlay with your incoming messages.
The nav drawer is displayed on the left edge of your browser display:

Note: you can also access the messaging preferences by clicking the gear icon at the top right of the messaging drawer.

Changing other notification preferences

To change other notification settings, including subscribed forum posts, click Notification Preferences from your Profile > Preferences area.

You can customize what to be notified about and how, based on your online status in eClass. Checking Disable notifications will silence all communications from eClass.

Choosing to receive notifications through eClass will enable the numeric visual indicator on your notifications icon:

 Clicking this icon will show a list of all your recent notifications:

Changing the email address that notifications get sent to

You could also change the email you get notifications sent to, if you don't want it sent to your UAlberta email. Go to Notification preferences in your Profile > Preferences area, and click on Email at the top of the list.

You can then add the email address that you want it sent it under "Send email notifications to". Click Save changes when done.