The court system of the United States, as defined by the Constitution, is divided into two branches; federal courts and state courts. Both types have specific areas of jurisdiction and certain types of cases they will hear. County and circuit courts fall under the state level but have different functions within the system.
State courts' powers are granted to them by the constitution. Amendment 10 clarifies that "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people." In other words, powers not specifically granted to the federal level will fall under the state level.
The 14th Amendment imparts certain restrictions on the abilities of state courts. This amendment was created to ensure that the rights of newly freed slaves were protected across the nation. It reads, in part, "No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States." State courts shall not create any laws that conflict with federal law.
The roles of the courts within each state may vary. States' court structures and jurisdictions are established by the state constitution. There are certain issues or types of cases that will always fall under state courts' jurisdiction. These include cases involving probate, election issues, family law and cases concerning the state constitution.
In Florida, for example, flcourts.org says circuit courts have jurisdiction in civil cases above $15,000, tax disputes, cases involving juveniles and criminal prosecutions for felonies. Florida has 20 judicial circuits with one court in each. Virginia, on the other hand, has 120 courts over the 31 judicial circuits. Like Florida, these are called courts of general jurisdiction. Virginia's circuit courts also hold jurisdiction over civil cases above $15,000, juveniles above 14, equity matters (divorces, wills, estates, etc) and trial of all felonies.
There are 67 county courts in Florida, one in each county. They have jurisdiction over civil cases of less than $15,000, traffic disputes and misdemeanors. Virginia's system does not have county courts per se, but rather district courts situated in each county. Virginia district courts hear cases involving misdemeanors, civil cases of less than $15,000 and traffic disputes. Virginia's district courts function in much the same way as Florida's county courts.
State constitutions will determine the makeup of their court systems, the number of courts and their locations, and the respective abilities and responsibilities of each. No two states have the exact same system.