The oldest courthouse in the United States is in King William County, Virginia. It was erected in 1725 – 50 years before the Revolution – and remains in use today. Likewise, the courthouse where American Patriot Patrick Henry served as an attorney and tried some landmark cases is still standing and in use in Hanover, Virginia, as it has been since 1737. In Illinois, a private residence that later served as a frontier courthouse in Cahokia still stands (but is no longer a courthouse) despite being constructed by “post-on-sill” style in 1740.
Architectural styles of courthouses vary but oftentimes they reflect Federalist style with tall columns flanking the front doors and wide hallways paved in marble to impress visitors with a sense of awe and decorum.
In some states the courthouse is located at the county seat, the geographical center of the county. In others you may encounter many courthouses, including city courts, municipal courts, and district courts at the local level. Within those buildings are many different divisions with a variety of responsibilities from collecting fines to settling disputes and issuing arrest warrants.
The curious are welcome to visit courthouses, but be prepared for a metal detector and guard at the gate these days. Just like airports and other secure facilities there are rules at courthouses about what can be carried in. Guards will usually allow you to set aside any inappropriate items like pocketknives to be picked up when you leave.
Courts can be repositories for more than records of legal proceedings: one may find family records, property records, and historical documents too.
Depending on the location, the clerk's office is a good first stop for finding historical and genealogical records. Generally the clerk's office schedules court business, but they are also helpful in finding birth, naturalization, local census, and marriage records and pointing researchers in the right direction for probate records. Many states will send you to the town clerk's office for the same records you may find in a court-based county clerk's office elsewhere. It's always a good idea to call ahead to find out what records are available as well as any restrictions on hours that those records may be accessed for research purposes. There may be a fee for copying records as well.
It's always worth checking for civil and criminal law cases that involved the person you may be researching: the documents describing such cases may include testimony from the individual and the case may describe a lot about the era when it happened, the type of business your subject was involved in, as well as what was at stake (whether it was a business dispute, a real property issue, or a small claims complaint).
Probate records show the disposition of assets when a person dies. These documents will list the family members of the deceased along with any information about property that was handed down or sold after the owner's death. Included in these files are wills that are likely to list all known relatives.