Facebook Messages won't work on OS X (10.9) Messages when two-step verification and other safety features are used on Facebook
You'll need to get an application specific password from Google.


Fire up your web browser and go to Google.com
Once there, sign in if you haven't already done so.
Go to the section where your picture is, and click on the "Account" link.
Once in that Account page, on the top there are links, click on the "Security" link.  Here you will probably notice that the 2-Step authentication is enabled.
Go to the App Password's "Settings" link. It will probably ask you to authenticate yourself, go ahead and do that.
Now you should be on the "Authorized Access to your Google Account" page, and you should be able to generate a new password for "Messages on Mavericks. Go ahead and give it a name, I named mine "Mac Pro Messages Google Talk Account", and then I clicked "Generate".
Once you have this password, go to "Messages" on your Mac, try to log in and copy/paste that password.


You should be all set.