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
The Holy Roman Emperor was widely perceived to rule by divine right, though he often contradicted or rivaled the Pope, most notably during the Investiture controversy. 
In theory, the Holy Roman Emperor was "primus inter pares" (first among equals) among other Catholic monarchs. In practice, a Holy Roman Emperor was only as strong as his army and alliances, including marriage alliances, made him. There was never a Holy Roman Empress regnant, though women such as Theophanu and Maria Theresa of Austria served as de facto Empresses regnant.
Throughout its history, the position was viewed as a defender of the Roman Catholic faith. Until the Reformation, the Emperor elect ("imperator electus") was required to be crowned by the Pope before assuming the imperial title. Charles V, Holy Roman Emperor was the last to be crowned by the Pope in 1530. Even after the Reformation, the elected Emperor always was a Roman Catholic. There were short periods in history when the electoral college was dominated by Protestants, and the electors usually voted in their own political interest.