Many county courts have searchable records online where you can see some details of a recently filed lawsuit (you can search by name usually and type of case). To find the online records, search for "circuit clerk mycounty" or something like that. The details available online might be just the docket sheet, i.e. this petition for relief was filed on this date and it is set for a hearing or whatever on this date. The case number is your ticket to look at the paperwork in the court file. You can go to the circuit clerk's office in the county court and ask to check out file number xyz.

If the records are not available online, go to the county courthouse to the circuit clerk's office. They might have a computer terminal where you can search to find the case number for the lawsuit you are looking for. Or just say you are looking for a civil case where xyz is a party. Once you have the case number, you can ask to see the court file.

All lawsuits or criminal cases are public record and available for anyone to review upon request unless a judge has ruled otherwise (or it's a juvenile case). You can make copies but probably have to pay per page.

If you're looking for the court file of an old case, it would probably be on microfiche, although some records are just destroyed after a certain time and a surprising number of records get destroyed by water or moisture in the courthouse basement or fire or are just lost. How long they keep what kinds of files is probably a function of local or state law.