DC has both local courts and federal courts.
LOCAL COURTS:
The DC Superior Court handles all local trial matters, including civil, criminal, family court, probate, tax, landlord-tenant, small claims, and traffic.
The DC Court of Appeals is the equivalent of a state supreme court. As the highest court for the District of Columbia, the Court of Appeals is authorized to review all final orders, judgments and specified interlocutory orders of the Superior Court of the District of Columbia.
To learn more about the local DC Courts visit the website https://www.dccourts.gov/.
FEDERAL COURTS
US Circuit Court of the District of Columbia. To learn more about the DC Circuit court visit the website
https://www.dcd.uscourts.gov/
US Court of Appeals for the DC Circuit. To learn more about the federal DC Courts visit the website
https://www.cadc.uscourts.gov/
US Bankruptcy Court learn more here: https://www.dcb.uscourts.gov/