Clause 1. The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, 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 Controversies 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 of different States, and between a State, or the Citizens thereof, and foreign States, Citizens or Subjects.
This section establishes the jurisdiction of the federal courts. Federal courts can decide cases involving federal law, disputes between different states, and disputes between residents of different states. (Cases involving state laws and disputes between residents of one state are heard in state courts.) The 11th Amendment, passed in 1795, curtailed federal jurisdiction in several types of cases involving state governments; that amendment accounts for the crossed-out bits here.
Clause 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. 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.
"Original jurisdiction" means the court hears a case in the first instance. "Appellate jurisdiction" means that the court hears a case only upon appeal of a previous decision from a lower court. The US Supreme Court has original jurisdiction in a few select kinds of cases—those involving ambassadors, for example. But in practice, the overwhelming majority of cases heard by the Supreme Court get there on appeal from lower federal courts or state courts.
Clause 3. The Trial of all Crimes, except in Cases of Impeachment, shall be by Jury; and such Trial shall be held in the State where the said Crimes shall have been committed; but when not committed within any State, the Trial shall be at such Place or Places as the Congress may by Law have directed.
Someone accused of a crime in federal court has the right to have his trial heard by a jury of his peers, rather than decided solely by a judge. The Framers of the Constitution believed that trial by jury was a crucial bulwark of liberty, preventing any potentially tyrannical judges from abusing their power.