Curious about the nomenclature of the empire and about the power balance between the two main constituent nationalities.
The "Habsburg monarchy" is generally suited to the time when the larger organizational nature of the hereditary lands ruled by the House of Austria was nebulous.
In 1804, with Napoléon in mind as a threat, the Holy Roman Emperor Francis II assumed the title (Hereditary) Emperor of the (House of) Austria. Aside from various territories being within the Holy Roman Empire itself, this was the initial concept of Austria as empire. The thing though is that the Austria this refers to wasn't a single geographical unit but rather the name of the ruling house itself.
The imperial title assumed in 1804 referred to the Kaiserthum Österreich (the 'Emperordom' of Austria, Austrian Empire), not the Kaiserreich Österreich (Empire of Austria). It was more or less a direct substitution of name for the informal term "Habsburg monarchy" which also established that the ruling house had and would retain imperial status, on a now-hereditary basis, since the writing on the wall indicated that the elective Holy Roman Empire was nearing its end. The imperial title represented all of the Habsburg hereditary lands, including the Kingdom of Hungary. The relationships between the territories were much the same as before until the uprisings of 1848-1849 when a constitution amalgamated Hungary under the Viennese administration. Until that point there wasn't even a common passport or citizenship for Austrian and Hungarian citizens.
The Compromise of 1867 split the Austrian Empire into two and established Austria as its own territorial empire (and not just of the ruling house) with the kingdom of Hungary and its associated territories no longer subject to Austria itself, but to be held as a real union under the Emperor of Austria as King of Hungary. This followed years of uprisings, attempted revolution, and anti-Habsburg and -German sentiment within the Hungarian/non-German-speaking lands because of rule by German-speaking/non-native bureaucrats. Greater autonomy and responsibility for their own affairs was a better way to keep the Hungarians happy and avoid losing the kingdom altogether. This territorial arrangement is referred to the synonymous terms "Austria-Hungary" or "the Austro-Hungarian empire".
After 1867, Hungary was responsible for its own internal governing. The military, navy, and some other institutions were shared.