In England and Wales we have a single legal system - which has criminal and civil aspects. In the United States the state legal system (for example that of Virginia, Maryland, Wisconsin or California) exists alongside the Federal legal system.
Virginia State Courts
"There is a general district court in each city and county in Virginia. The general district court handles traffic violations, hears minor criminal cases known as misdemeanors, and conducts preliminary hearings for more serious criminal cases called felonies.
General district courts have exclusive authority to hear civil cases with claims of $4,500 or less and share authority with the circuit courts to hear cases with claims between $4,500 and $15,000. Examples of civil cases are landlord and tenant disputes, contract disputes and personal injury actions."
"The circuit court is the trial court of general jurisdiction in Virginia; this means that the court has authority to try a full range of cases both civil and criminal. Civil cases involve disputes essentially private in nature between two or more parties; criminal cases are controversies between the State and persons accused of a crime. Only in a circuit court is a jury provided for the trial of these disputes and controversies."
A diagram of the Virginia State Court structure is accessible here.
Federal Courts
"The United States district courts are the trial courts of the federal court system. Within limits set by Congress and the Constitution, the district courts have jurisdiction to hear nearly all categories of federal cases, including both civil and criminal matters. Every day hundreds of people across the nation are selected for jury duty and help decide some of these cases.
There are 94 federal judicial districts, including at least one district in each state, the District of Columbia and Puerto Rico. Three territories of the United States -- the Virgin Islands, Guam, and the Northern Mariana Islands -- have district courts that hear federal cases, including bankruptcy cases."
A diagram of the structure is available here.
So in Alexandria, Virginia you will find
STATE
The Alexandria General District Court at 520 King Street
The Alexandria Circuit Court at 520 King Street
(The Court of Appeals and the Virginia Supreme Court are 100 miles away in Richmond)
FEDERAL
Virginia Eastern District Court at 401 Courthouse Square
Virginia Eastern Bankruptcy Court at 200 South Washington Street