Re: ...it generally means a trial court, or a court of first resort. This is where facts are presented, as opposed to appellate courts, where the court will only review issues of law.
With NY state being an exception, here the "Court of Appeals" is the court of first resort! One then goes to the Appellate Division of the Supreme Court, and finally to the Supreme Court.

anyone who watches any of the variations of the show LAW & ORDER, has seen the entry of NY's Supreme court, segments of the show are filmed on the courthouse steps. the site has been used in many movies as well.
(the interior shots are mostly filled in the Appellate Division court house several miles away, one on NYC most beautiful state court houses. The same location is used for swearing in new members of the bar. the background murals are quite beautiful. The interior for the show are filmed in close by building to the actual DA office, (the background building indicate that) but the real DA's office is on the 10th floor, and only has a view of other buildings, not of the skyline and Brooklyn Bridge .