Moderator actions are invoked using the moderator bar that is at the top of every page. In this post, I'll be focusing on how moderators handle reported content.
Handling a report is a little more formalized than you might be used to. Reports are now tracked with a status, so that it's clear what has happened and, if you leave a comment, why.
This bar makes it easy to see when you have tasks outstanding, but doesn't get in the way of reading the forum.
Inside the reported items section, you are presented with a list of outstanding (open or assigned) reports:
This does not just have to show a list of reported posts. It will show any type of comment, including profile posts or those added by add-ons. (Add-ons simply need to implement the basic reporting functionality and write some code to interact with this system.)
You can also use the button in the upper right to view recently closed reports.
Diving into the report listed, we see:
This shows us the content that was reported and any comments that may have been made. If a piece of content is reported multiple times, only one report will be created but all the comments will be listed here. Each moderator that can manage the report has the option of making a comment as well. This is important when the type of action that needs to be taken isn't 100% clear. Note that users that report content can't see this page or any of the comments that are made; they are for internal use only.
To modify the report's status, you must first claim the report. This is mostly important if you have two or more people who might handle the report. Once one person claims the report, the other can see that it's being handled. After claiming the report, the view switches slightly:
Now, using the "go to content" button (which opens in a new window), you can manage the content directly. After you have taken your course of action, you'd come back to the report view and change the status. After submitting, you're taken back to the report list view to move on to the next thing.
After a report has been closed, it can be re-opened if needed:
A moderator's comment will automatically re-open it. Another user reporting the content will cause it to be re-opened; however, all of the comments will still be available to help you decide whether an action is really necessary.
Added September 22nd:
Floris has put together a video showing how the system works. Thanks, Floris!