The permit you are talking about is a Recreational Passport. It costs $11.00 for the year for residents and $31.00 for non residents. There is also a daily permit avalialible to non residents for $9.00 a day.
These permits allow you access to all Michigan Stata Parks, Boating Access facilities, State Forest Campgrounds. Camping fees I believe start at $25.00 for electric only sites up to around $35.00 for water sewer and electric sites. Not many parks offer full hook up sites but most have a dump station and they all have a 14 day stay limit.