They hear cases that violate your US Constitutional rights.
The US District Courts hear approximately 80% of new federal cases; the US Court of International Trade and US Special Courts hear the remainder of the cases under original jurisdiction.
This is admittedly an educated guess and may be applicable only in the US: But I believe the answer would be Municipal Courts or County-level court systems probably hear the most cases due simply to the sheer number of misdemeanor and traffic cases that are heard each day.
The US Supreme Court hears such cases every year.
The Supreme Court justices hear cases in the courtroom of the Supreme Court Building in Washington, DC.
Yes, the Court picks and chooses the cases that it will consider. Refusal to hear a case does not have to be explained to the satisfaction of the participants.
Supreme Court
The US District Court of the jurisdiction in which the offense occurred.
US Special Courts or courts of special jurisdiction hear a variety of cases, each related to a particular subject matter. For example, US Bankruptcy Courts hear bankruptcy cases; US Tax Court hears federal tax cases; the US Court of Federal Claims hears cases involving monetary damages against the US government; the US Court of Appeals for Veteran's Claims hears appeals of decisions made by the Board of Veteran's Appeals, and so on. Military courts are also considered special courts.
The US Supreme Court typically hears oral arguments on two, sometimes three, cases per day, three days per week (currently Monday, Tuesday and Wednesday). At this rate, the Court hears approximately 12-18 cases in a two-week sitting.
When judges traveled from place to place to hear cases, it was called "circuit riding."
Yes. US District Courts are the trial courts of general jurisdiction. They hear civil and criminal cases that involve federal and constitutional law and US treaties, provided the case doesn't fall under the jurisdiction of one of the US Special Courts (bankruptcy, tax, etc.).