Ok,
I'm still learning _exactly_ how this works, but I've got a pretty good feel for it.
You'll notice on the main page, when there is a new message, the blue box turns from

to

.
Blue means there hasn't been any activity since you've been in that forum. Yellow means there are new messages you haven't seen since the last time you were active on the forums.
So now you're wondering, what happens if there are 12 posts on a particular thread, and you've read 5-8 of them, but aren't sure. Look for this icon next to each post:

. It means this is a new message.

means you've read it already.
Once you've read the messages, and after 15 minutes of inactivity, the yellow icons will turn back to the normal blue.
I'm pretty sure this will work on multiple computers, because the board keeps track of the last time each user has been active (view, post, etc) on the forums.
Hope that helped.