Facebook Insights is to your Facebook page like Google Analytics is to your blog. Insights tells you who's visiting, who's talking and other in-depth information that will (hopefully) help you optimize your page and drive more traffic to your blog.
To access Facebook Insights, login to Facebook, go to your page and click the "View Insights" link in the sidebar:
There's a new version of Insights available so you might see the following screen. Click "Take me to the new Page Insights" link here:
As far as reading your Insights and interpreting them, I am unabashedly cheating in this post and sending you elsewhere. Because I think it's goofy to reinvent the wheel.
There's a post up at Income Diary that breaks down Facebook Insights in a detailed way: The Complete Guide to the New Facebook Insights.
There's helpful stuff there. If you want to optimize your Facebook use (or if you're just curious about who your Facebook visitors are), check it out.
Here at BloggingWithAmy you'll find tips, tools & tutorials about blogging and websites. In plain English. Want to 
Oh, and I thought FB profile visits are confidential info
I am sure there is more to this than profile visits information. Is this available for all users?
Facebook Insights is most helpful info your Facebook page (as opposed to your profile).
Okay, now I'm obsessed with this!!!
One thing I find odd and am trying to understand is that I had fewer page views on the day that I had the most comments ever on a post (100) than the next day when I just had a handful of comments. Maybe people were checking back to see what the comments were???
Also interesting to see that I got the most "unlikes" when I posted 2 anti-abortion links. (Some things are worth being "unliked" for, though!)
Amy…any idea why one page of mine said "old insights" only and I can't switch?
This page also won't allow me to change which tab is the one that displays on landing (so I can't use a custom welcome page).
I've taken the advice of some Social Media Gurus and focused on only social media until you're very comfortable with it and feel like you're part of a community before moving on.
Been very lame about my Facebook Page but knowing and seeing stats like this make it even more interesting, to see who's coming and from where. Gives you deeper insight on what's going on there beyond seeing people who comment.
Thank you for the referral!
Thanks for this advice Amy! It really is helpful to see the activity behind the scenes so we can understand and link together what perhaps is working and not. Awesome!
I went to Facebook and could not find any "insights" link at all. I even put it into my "find" in IE, and it could not find it, either. Do I need to have a business-type page for this option to be available? Thank you.
Hi Shirley. Yes the Insights would be found on your Page (as opposed to your Personal Profile).