Beside each forum category is a red folder.

When there are no new posts, the folder will look exactly like the one above. A little faded and not as bright. Like so:

When there is a new post, the red folder will be more vivid and far more red in color, than the previous pictured folder. Like so: 

And that's how you can tell if there's been any new posts or activity within a forum.**

**This doesn't seem to be the case for Child Boards, unfortunately.