Marv's answer is pretty much on the mark. But there's a caveat. It depends on which year your ancestor was naturalized. Prior to the Immigration reform of 1905, the requirements for naturalization were...., well for lack of a better word, they were lax.
Most records are filmed and available through both the Library of Michigan and the National Archives and Records Administration in Chicago. You're free to go browse and make copies. But there are isolated pockets of records that haven't made their way there. The reason is that there were several different places a person could go to take the Oath of Allegiance. One box of records from the Federal Courthouse in Bay City were sent to Detroit and stay there. No one knows why they haven't gone on to NARA, but they haven't. There's a list of those at the Federal Court in Bay City.
There were also two boxes of records from a Justice of the Peace in Bay City that simply disappeared in 1978. No one has any clue where they ended up, but they're records from 1888-1895 and even the list of the records was lost.
As for the information you need if you write to NARA or the State requesting a copy of the records, you need to provide the name, date of birth, date of naturalization (as close asyou can come) and the town where you believe they were naturalized...or at least the area of the state.
The records are very disorganized, so if you have the ability to go to Lansing or Chicago to search the films in person, you'll probably have better luck than asking someone else to do the lookups for you. They were organized and filmed by box in many instances, so you have to go through film after film...and hope that none was misfiled and out of order.