Emperor
The first Austrian Emperor was the last Holy Roman Emperor Francis II. In the face of aggressions by Napoleon, Francis feared for the future of the Holy Roman Empire. He wished to maintain his and his family's Imperial status in the event that the Holy Roman Empire should be dissolved, as it indeed was in 1806 when an Austrian-led army suffered a humiliating defeat at the Battle of Austerlitz. After which, the victorious Napoleon proceeded to dismantle the old "Reich" by severing a good portion from the empire and turning it into a separate Confederation of the Rhine. With the size of his imperial realm significantly reduced, Francis II, "Holy Roman Emperor" became Francis I, "Emperor of Austria". The new imperial title may have sounded less prestigious than the old one, but Francis' dynasty continued to rule from Austria and a Habsburg monarch was still an emperor ("Kaiser"), and not just merely a king ("König"), in name.
Holy Roman Emperor
While earlier Germanic and Italian monarchs had been crowned as Roman Emperors, the actual Holy Roman Empire is usually considered to have begun with the crowning of the Saxon king Otto I. It was officially an elective position, though at times it ran in families, notably the four generations of the Salian dynasty in the 11th century. From the end of the Salian dynasty through the middle 15th century, the Emperors drew from many different German dynasties, and it was rare for the throne to pass from father to son. That changed with the ascension of the Austrian House of Habsburg, as an unbroken line of Habsburgs would hold the Imperial throne until the 18th century, later a cadet branch known as the House of Habsburg-Lorraine would likewise pass it from father to son until the abolition of the Empire in 1806. Notably, the Habsburgs also dispensed with the requirement that emperors be crowned by the pope before exercising their office. Starting with Ferdinand I, all successive Emperors forwent the traditional coronation.