The courthouses of America have many different types of records to aid patrons in their search. Vital Records are often kept there, as well as well, property records, probate records, and records of court proceedings.
Many clerks even let citizens search the records themselves.
Other courthouses are very formal and insist on looking everything up themselves, for a price. Some will let you make copies of pertinent family records. Each is courthouse is different, based on local laws, and will have to be dealt with individually. Many courthouses have agreed to let LDS microfilmers in to film their old records. If so, you may order copies of these records at your local LDS Family History Center.