If you would like to see which forums have new posts in them, you can see this in any forum or subforum as the icon for that forum turns blue when there are new posts in it.

If you see a post listing, there can be one or more icons in it. The blue icon (if present) will take you to the first message in that forum that you have not yet read. The icon beside it will take you to the very last message in that forum.

View Unanswered Posts
The "View Unanswered Posts" icon in the upper right will present you with a list of all messages that have no reply. This is helpful if you are looking to help people and want to see if anyone has not been answered yet.

View Unread Posts
The "View Unread Posts" icon in the upper right will present you with a list of all messages that you have not yet read. This is on the entire forum.

View New Posts
The "View New Posts" icon in the upper right will show you all posts in the last "X" amount of time. I'm not sure what "X" is - that's probably an admin setting.

View Active Topics
The "View Active Topics" icon in the upper right will present you with a list of all topics that meet the definition of "active" set by the forum. This is probably based on number of posts and how recent they are.

View Your Posts
Self-explanatory.

Mark Forums Read
"Mark Forums Read" will do just that - it will mark all forum posts as if you went in and read them. This works with "View Unread Posts" to help you find only posts since your last visit.

Screen shots later - not possible on this computer.