What do you call it?

Possibly a circuit court?

circuit court?
Can we call such a circuit court a permanent circuit court based in a sub-district?

If it's permanent, you might want District Court.

in florida we have:
supreme court, which is over all district courts of appeal, which are over 'circuit' and 'county' courts

the 'main' court in your statement (as far as florida is concerned) would be the 'surpreme' court, with 2 levels of Courts beneath it.

There are different names for American courts, depending upon the state. Usually, District Court is lowest (Small Claims Court is actually lower, but it doesn't have jury trials so it's probably not what you're thinking of), then Superior Court, then Appeals Court (or Appellate Court), then Supreme Court.

However, in New York it's Supreme Court, then Appellate Court, then the Court of Appeals (yes, the Supreme Court is the lowest one in New York). In Massachusetts, it's Superior Court, then Appeals Court (I believe, it's been a while since I had to know this), then finally the highest court is the Supreme Judicial Court. Louisiana has Parish Courts. Delaware has an Equity System. It's all over the place.

In the American Federal system, things are confused even further. Then it's District Court, Circuit Court and then finally the US Supreme Court.

Of course, in England, Canada, France, etc. the system is different and so the courts have differing names.

You can't go wrong by saying more generalized terms like lowest court, intermediate appeals court and highest court, but if you want to be accurate and precise, you really should check what it is, depending upon which area you're talking about.

Jespah's right: it's all over the place in the US. In Illinois, trial courts are circuit courts and appeals courts are district courts. In the federal system, it's just the opposite: trial courts are district courts and appeals courts are circuit courts. Some states still have JP (justice of the peace) courts. Delaware and Tennessee have separate chancery court systems. Texas has separate appellate courts for civil and criminal appeals. Some states don't have intermediate appellate courts at all (e.g. New Hampshire, Wyoming). It's all very confusing.