Boards show little green ‘pips’ at the bottom right corner when they have unseen activity, so it's easy to see at a glance what boards have new activity, and how much new activity they have.
Click on the pip to get an overview of the unseen activity, and if the board has sub-boards you'll be able to see the activity breakdown for each. You can then jump to a sub-board, or clear the activity by clicking 'Mark all as seen'.
You can also click the activity summary pip at the top to filter to just the boards that have unseen activity.
If you go to 'Sort by' and hit 'Activity count', you will see your boards with the most unseen activity first.