Private messages (PMs) is a new feature of the new software. It's sort of a forum based email, but instead of sending via email, it sends it via the forums. When you have unread (new) PMs, you'll get the pop-up notification. You can either click on the pop-up to go to your inbox, or go there from the bottom-left of each forum page as Duncan says. Once you've read all the messages in your inbox, the pop-up will go away.
Under each member's name and avatar on each post is a "Send Message" link. Use this to send PMs to other members.