This from the Washington Post this morning...
There are a whole bunch of key government functions that carry on during a shutdown, including anything related to national security, public safety, or programs written into permanent law (like Social Security).
Here's a partial list:
Any employee who conducts "essential activities to the extent that they protect life and property." So, for example: Air traffic control stays open. So does all emergency medical care, border patrol, federal prisons, most law enforcement, emergency and disaster assistance, overseeing the banking system, operating the power grid, and guarding federal property.