Parking downtown, around campus, and in the neighborhoods that adjoin those areas is one of the frequent complaints people have about Ann Arbor. In general, though, parking can be found downtown and near campus at nearly any time of day or night - the parking structures cost a mere $1.50/hour, and only a few of them ever fill up completely. Free downtown parking, like a free lunch, is harder to find, though it does exist if you are in the know or willing to walk a bit.
Free downtown parking
In general, all parking spaces in the Ann Arbor downtown are reserved, metered, or restricted to customer or commercial delivery use during daytime business hours.
After 6 pm, some loading zones are free parking.
After bank business hours, some bank lots are free parking.
After county business hours and on weekends, county lots are free parking. Note that city lots for city employees are not free.
On Sundays, city structures are free.
Neighborhoods without a posted "no parking" sign are also fair game. Some neighborhoods restrict street parking to 2 hours for non-residents, but it's still free.
The DDA used to provide a concise real time update of parking availability at http://a2dda.org/parking.php suitable for use on cell phones; that page now generates a 404 error. The data that was there is now at http://www.a2dda.org/parking__transportation/available_parking_spots/ . The table format on the new page has a colon after each parking structure name, but otherwise is very similar to the table format on the old concise page. AAPark makes this data available via text messaging and can also text you back when the structure you're interested in is getting full.
University of Michigan
The University maintains a large quantity of permit parking and some hourly parking around campus: http://www.pts.umich.edu/
In some of the near-downtown neighborhoods, a Residential Permit Parking program is in effect, with a 2 hour limit during the day on vehicles without a sticker. This program has bred some controversy, with some stating that on-street parking should be unregulated, others that it should be both reserved for residents and free, others saying it should be more expensive (the program costs the city money), and many interested just in tweaking the formula and process. The RPP has proved a source of much discussion on Ann Arbor is Overrated and ArborUpdate. The City's information on RPPs is at: http://www.a2gov.org/government/publicservices/customerservice/Pages/ResidentialParkingPermits.aspx