How many courts and judges are in the Supreme Court? Also, what types of cases do they receive?

While the jurisdiction of the United States Supreme Court is largely appellate, it does have what is called original jurisdiction in a few instances. This means that the Supreme Court acts as a trial court, taking evidence as any trial court does.  This jurisdiction is set forth in the second paragraph of Article III, Section 2:

In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court shall have original jurisdiction.

This does not come up very often, but the Constitution makes clear that if an issue arises over certain kinds of public officials who are involved in international diplomacy, the Supreme Court has original jurisdiction.  Similarly, disputes between states fall under the original jurisdiction of this court.  I have included two links to two different cases in which the court had original jurisdiction over the states as parties. These are probably the most recent examples.