Short answers, pulled from the story.
Charlemagne, King of the Franks, became the first Holy Roman Emperor when Pope Leo III crowned him in Rome on the 25th of December 800. An emperor already reigned in Constantinople at the time, and the coronation invoked the concept of translatio imperii to transfer that authority westward.
The Holy Roman Emperor was the ruler and head of state of the Holy Roman Empire, a title the Catholic Church treated as the sole successor of the Roman Empire. Diplomats described the emperor as primus inter pares, first among equals, among Europe's Catholic monarchs.
Holy Roman Emperors were elected by a body of prince-electors rather than inheriting the office automatically. The Golden Bull of 1356 named seven of them: the archbishops of Mainz, Trier, and Cologne, the king of Bohemia, the count palatine of the Rhine, the duke of Saxony, and the margrave of Brandenburg.
Tradition held that no king could call himself emperor without a coronation performed by the pope in Rome. That requirement changed in 1508, when Pope Julius II let Maximilian I take the title without traveling to Rome, and Charles V became the last emperor ever crowned by a pope, in Bologna in 1530.
The Holy Roman Empire ended in 1806, when Francis II dissolved it after a devastating defeat by Napoleon at the Battle of Austerlitz. Francis II, the last Holy Roman Emperor, already held a separate title as Emperor of Austria.
The House of Habsburg held the Holy Roman Emperor title almost continuously from 1452 to 1806, with only one interruption. Its cadet branch, the House of Habsburg-Lorraine, held the title for the final stretch, from 1765 to 1806.