Unfortunately, this is a little bit hidden in the current UI. Here's how you enable desktop notifications:
- Go to a board
- Scroll to the bottom of the "Activity" widget (lower right)
- Click "Allow Notifications"

This setting is per-browser (so you'll need to re-enable it if you use a different computer), and only works in Chrome.
If you don't see that link, check the Notifications section of chrome://settings/content
to see what your notifications settings are (they may be set to "Do not allow"), and chrome://settings/contentExceptions#notifications
to see which sites you're currently allowing/preventing notifications on.
Notes
- You'll only get notifications when someone @mentions you in a comment (e.g. when they say "Hey @Derek, what do you think?")