how does the supreme court choose which cases to take? who may appeal to them?

Answer 1

“The judicial Power shall extend to all Cases, in Law and Equity, arising under thisConstitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;—to all Cases affecting Ambassadors, other public Ministers and Consuls;—to all Cases of admiralty and maritime Jurisdiction;—to Contro- versies to which the United States shall be a Party;—to Controversies between two or more States;—between a State and Citizens of another State;—between Citizens of different States;—between Citizens of the same State claiming Lands under Grants ofdifferent States, and between a State, or the Citizens thereof, and foreign States, Citizensor Subjects.
“In all Cases affecting Ambassadors, other public ministers and Consuls, and those inwhich a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.”
Appellate jurisdiction has been conferred upon the Supreme Court by various statutes,under the authority given Congress by the Constitution. The basic statute effective at this time in conferring and controlling jurisdiction of the Supreme Court may be found in 28 U. S. C. §1251 et seq., and various special statutes.

Answer 2

Cases reach the Supreme Court when the Constitution must be interprested to uphold the law or the case has gone through all other courts. The Supreme Court is the end of the line, though -- no one can appeal one the Supreme Court has made a ruling.