Any federally-funded program not deemed "necessary for the safety of life and protection of property" is shut down, though there appears to be no well-defined definition of this set. Most essential services (e.g. law enforcement) are covered under this definition, and many others have existing money to tide them through a shutdown (e.g. Medicare and food stamps). I was able to identify only a handful of changes likely to immediately affect many people:
- Paper-based tax refunds and all tax audits would be halted, which could be substantive at certain times of the year 
- The Federal Housing Administration would stop processing loan guarantees: "A shutdown of just a few days would cause some inconvenience and delay to FHA mortgage applicants looking to close on a new house -- as well as private lenders unsure of whether to approve loans that normally need to be endorsed by the federal government." 
- Government employees put on furlough, most likely without pay. This won't effect most Americans, obviously, but "A senior administration official said a shutdown could idle 800,000 of the 1.9 million civilian federal workers." 
- No new passports or visas issued 
What is frustrating for government employees on the eve of a possible shutdown is that many have no idea whether their jobs are deemed exempt from a shutdown, says John Gage, president of the American Federation of Government Employees, the largest federal employee union.
With the possibility of a government shutdown looming, Obama administration officials raced Wednesday to identify essential government services that would have to be provided if the government ran out of money. 
The NYTimes article  is the most comprehensive I'd found.
More discussion on 2013 U.S. Federal Government Shutdown:
- Spotlight on the 2013 U.S. Government Shutdown
- Should the Republicans have worked harder to socialize their shutdown strategy with the broader American public?
- What are some plausible endgame scenarios to the budget standoff?