The Habsburg Empire was named after Habsburg Castle (also known as Habichtsburg) overlooking the Aar River. The castle was built in 1020 by Count Radbot and his brother-in-law Werner. Radbot’s son became the first Count of Habsburg. He was the grandfather of Albert III, who was Count of Zurich and the landgrave of Upper Alsace. (A landgrave was a count with jurisdiction over a particular territory.) After the death of Albert III and of Rudolf II of Habsburg the associated territories were divided between Albert IV and Rudolf III; however, the descendants of Rudolf III sold their territory back to the descendants of Albert IV. It was his son Rudolf IV who became German king in 1273, although this meant that rather than being known as Rudolf IV he became known as Rudolf I. His sons Albert (who would become Albert I) and Rudolf (Rudolf II of Austria) inherited land including most of Austria as their territory.
Rudolf II renounced his share in the territories but, after the death of Albert I, there were problems with how the territory should be divided. The problems were eventually settled, with Rudolf IV taking the senior position in a partition of the territory with his brothers Albert III and Leopold III.
By this time Rudolf III was ruling Austria and Frederick I had become King of Germany (as Frederick III) while Albert V of Austria became German king (as Albert II). His son Ladislaus Posthumus also inherited the crown of Hungary in 1446. He was the last male descendant of Albert III. At this point the line of descent from Leopold III had divided into two surviving branches, the Inner Austrian branch and the Tyrolean branch. Frederick V was the senior member of the former and he now became German king, though he was known as Frederick III rather than Frederick V.
One of the Frederick’s first acts was to ratify the Habsburgs’ use of the title of ‘Archduke of Austria’ (which had earlier been arrogated for or to them by Rudolf IV). His son Maximilian united the Austrian hereditary lands when Sigismund of Tyrol abdicated in his favour (in 1490).
Maximilian married Mary, the heiress of Burgundy, which meant that their son Philip would inherit the territories of Charles the Bold, including Artois, the Netherlands, Luxembourg and the County of Burgundy or Franche-Comté. Philip went on to marry Joan, heiress of Castile and Aragon as well as other places such as Naples, Sicily and Sardinia. Maximilian was succeeded in the role of Habsburg monarch by Charles V.
It is important to be precise about how we talk about the Habsburgs as there are various different nomenclatures that can be applied to the dynasty. The Habsburg Monarchy (Habsburgermonarchie) is essentially a way of referring to monarchy that had a descendant of the Habsburgs as its titular head. This is contrasted with the Habsburg Empire (Habsburgerreich) which is the empire ruled over by the descendants of the Habsburgs.
The core part of the Habsburg dynasty’s territory that was ruled over directly by the Habsburg monarchy was known as either the Habsburg Hereditary Lands or Austrian Hereditary Lands (Habsburgische Erblande or Österreichische Erblande). These are to be contrasted with the Austrian Monarchy (Österreichische Monarchie) and also with other monarchically ruled areas. In later centuries other pieces of nomenclature were used. For instance, the Austrian Empire (Kaisertum Österreich) could refer to the empire ruled from Austria or to a ‘widespreading domain’ thereof. Austria–Hungary would become the official name of the monarchy’s territory – this was also sometimes referred to as the Double Monarchy (Doppel-Monarchie) or, colloquially, as the Danubian Monarchy (Donaumonarchie).