In general: all “nonessential” governmental functions will be shut down. This could affect your ability to obtain information from some of the informational “800” numbers operated by the federal government.
If you have filed for a passport renewal, or a visa you will most likely be affected.
If you are seeking information from a federal agency (such as social security adminstration) or need information or documentation regarding a federal pension, you may face longer than usual wait times. This could indirectly affect your ability to complete an inventory and appraisal in a divorce or probate case. FOIA (Freedom of Information Act Requests) could have additional time added on to their already slow process times.
Social security benefits, applications and investigations would be delayed.
Many governmental employees will be affected. In the past, some have received pay later, while some people who work under federal contract have had pay suspended.
At this time it is uncertain whether a shutdown will occur, or how long one would last if it did.