Not sure what could have caused that. We haven't heard of any database errors before that cause the site to not work. Most occur during installation and are due to unwritable directories.
As for your current dilemma, if you run the following queries on your database, it should disable all of the JFBConnect plugins. Hopefully, that will allow you to get into the admin area again.
UPDATE xxx_extensions SET enabled = 0 WHERE element = "jfbcsystem";
UPDATE xxx_extensions SET enabled = 0 WHERE element = "jfbconnectuser";
UPDATE xxx_extensions SET enabled = 0 WHERE element = "jfbconnectauth";
Change the xxx in each statement to your database prefix.
After you're back up, a couple of questions:
Do you know what version you were using before you upgraded?
What wasn't working before today (with the old version)? That will help narrow down the original cause.
Sorry for the trouble, but we'll get you going,
Alex