District courts are the federal equivalent of state trial courts. Court of Appeals for XX Circuit is the equivalent of the state appellate court.
State: Trial court (Superior Court in MA, Supreme Court, confusingly, in NY, but your general, "Law and Order" trial court.)
Federal: District Court (e.g. District Court for the Southern District of NY is the Manhattan federal trial court; District Court for the Eastern District of Pennsylvania is the Philadelphia-and-environs federal trial court)
State: court of appeals (name varies sometimes, though)
Federal: Court of Appeals for the XX Circuit (aka just "The First Circuit" or "5th Circuit"--New England and much of the South, respectively)
Supreme Court level:
State: State Supreme Court (e.g. Massachusetts Supreme Judicial Court, Pennsylvania Supreme Court--confusingly, in NY it's called the New York Court of Appeals)
Federal: US Supreme Court
Federal district courts are seen as less prestigious because they still do trials, and not schmancy appellate work (except in rare cases like habeas). CoA (circuit courts) are more prestigious. But a really good fed district court, like the Southern District of New York, is probably just as prestigious as one of the more overlooked Circuit Courts, like the 10th Circuit or something (ID, CO, some other Rocky Mountain states, I think). It's generally an east coast bias thing, like so much else.