Short answers, pulled from the story.
King of the Romans was the title held by a German king between his election by the princes and his coronation as Holy Roman Emperor by the Pope in Rome. From the 16th century onwards, it was used solely for an elected heir during his predecessor's lifetime, after German rulers stopped seeking Papal coronation.
Henry II was the first to hold the title in its formal sense, from 1002. The title had roots in earlier Ottonian practice, when the king was styled King of the Franks, but the specific designation King of the Romans took hold from the late Salian period.
Henry IV adopted the title Romanorum Rex to press his sacred claim to be crowned Emperor by the Pope, signaling that his authority extended beyond Germany to the whole Empire. Pope Gregory VII deliberately countered this by calling him Teutonicorum Rex, King of the Germans, to imply his power was only local. Henry was finally crowned Emperor in 1084 by Antipope Clement III.
Under the Golden Bull of 1356, issued by Emperor Charles IV, only seven Prince-electors could vote: the Prince-Archbishops of Mainz, Trier, and Cologne, plus the King of Bohemia, the Count Palatine of the Rhine, the Saxon duke, and the Margrave of Brandenburg. Earlier practice had allowed all present noblemen to vote by acclamation before the franchise was narrowed. The Golden Bull remained in force until the Empire's dissolution in 1806.
Charles V was the last king to receive the Imperial Crown from a Pope, crowned by Pope Clement VII in Bologna in 1530. After that, rulers of the Empire styled themselves Emperor upon coronation in Germany without seeking Papal approval, beginning with Ferdinand I.
The title effectively ceased with the Empire's dissolution in 1806. Napoleon I introduced a related title, King of Rome, for his son Napoleon II at birth in 1811, though from 1818 onwards the boy was officially styled Duke of Reichstadt by his grandfather, Emperor Francis I of Austria, the last Holy Roman Emperor.