a government building that houses the offices of a county government
a building that houses judicial courts
a "nonpublic forum," which is to say a piece of property that's owned by the government, and that hasn't been opened as a place for people to speak to the public
a public building, one that the people own and one that the people occasionally HAVE to visit
A courthouse (sometimes spelled court house) is a building that is home to a local court of law and often the regional county government as well, although this is not the case in some larger cities. The term is common in North America. In some other English speaking countries buildings which house courts are simply called "courts".