The ability to have private conversations on Facebook through your private message inbox is of little importance if that inbox won't open. Logging out of Facebook and signing back in can cure some temporary glitches, but if that's not having any effect on opening a stubborn inbox, you'll need to do some additional troubleshooting.
Websites generally save cookies onto your computer when you visit to make the site load faster the next time you return. Sometimes these cookies can cause problems with site performance, so clearing them from your browser can help improve this and restore functionality on other sites that are misperforming. How to clear these cookies varies depending on your browser, but you can usually find a way to remove them, and otherwise clear your cache, under the Settings or Tools menu (see Resources). Updating your browser to the newest version may help correct functionality glitches like a stubborn inbox as well.
If the problem is originating from within Facebook itself, it's either a problem with the site or with your account. Facebook has the occasional glitch from time to time that can undermine account performance, in which case it's usually resolved within a few hours. Try opening your inbox later in the day to see if your stubborn inbox is resolved. If the issue does not correct itself in a day or so, it might be a problem with your account. In this case, contact Facebook though their Help Center and alert them to your issue.