Historiography on the Habsburg Realms and Austrian Empire was heavily influenced for a long time by its collapse at the end of the First World War, which characterized it as a regressive regime and the "prison house of nations." Neither of these assertions are true, which has been demonstrated by the recent scholarship on the Austrian Empire. The Habsburg Empire: A New History by Pieter M. Judson is a great example of this new generation of Habsburg historians. If you want a sense of the older rendition, see Robert A. Kann, A History of the Habsburg Empire, 1526-1918 (this is still the best overview history of the empire despite the caveats).

Until the dissolution of the Holy Roman Empire, the Habsburgs led the German states and had significant sway over Catholic europe, especially after the Spanish Habsburgs died off in 1700 and the Austrian branch eclipsed them. The empire adapted and reformed, which is all the more admirable given the linguistic and ethnic diversity that the Eastern Habsburg lands encompassed. While the early modern Habsburg empire was more a collection of territories, Maria Theresa and her son Joseph II made it into a much more unified state. Austria was also a major player in the Napoleonic Wars and the prime minister, Metternich, started the Congress of Vienna and Congress System that helped maintain peace in Europe for most of the century. While there were brief experiments with what could be called "absolutism" such as under Baron Alexander von Bach, this proved to be extremely unpopular.

The empire did suffer a hit to their international power because of Franz Joseph's handling of the Crimean War* and the rise of the North German Confederation/German Empire during the middle 19th century. Even with this, they were still an important part of continental balance of power.

The Austrian Empire is poorly understood in most US school history textbooks because it was a polyglot empire, and it was supposed that all of these national groups wanted their own independent countries. This was not the case until the final years of World War One. Most national groups wanted increased rights and privileges in the empire, and something along the lines of "federalism" as we have in the United States, although adapted to the empire's circumstances. Many of the older histories of the Austrian Empire were also motivated by nationalism, as the countries that were created after World War I had to justify their existence.

*See Paul W. Schroeder, Austria, Great Britain and the Crimean War: the Destruction of the European Concert

Other comments in this thread have gone into detail into the historiographical reasons why Austria has gotten the short end of the stick in terms of the common historical wisdom. Espescially in Anglophone history, there’s a strong tendency to think of the Austrian/Austro-Hungarian Empire as some kind of anachronism; a medieval kingdom that had somehow survived to the modern world and inevitably doomed to finally succumb to the forces of modernity. I’ve written a bit previously on the same factors affecting Austrian military accomplishments, and I would wager that some of the factors there also apply here.

While the Austrian Empire of late modernity is undoubtedly an interesting case to study, its outside of my forte so I’ll try and focus more on the early modern Habsburg state, prior to the French Revolutionary and Napoleonic Wars. Within the Holy Roman Empire (HRE), Austria did stand as primus inter pares amongst the German princes, as recognised by the continual election of Habsburg Austrian Archdukes as Emperor. Although the Imperial Diet could vote money to the Emperor, it was expected that the Emperor would mostly pay for the troops needed to defend the HRE out of their own wealth. With their control of Austria, and later Bohemia and Royal Hungary, the Habsburg were probably the best candidate in the HRE for this, even more so given that they were well positioned to face the apparent biggest threat—the Ottomans.

Although our modern view of the HRE is strongly shaped by the image of it as an ineffectual organisation, the position of Emperor did command a great degree of power and respect in European politics at the time. We tend to discount this looking back since ultimately the German Empire was founded under Prussian leadership, but for much of the early modern Austria was the preeminent power in central Europe. With the Great Turkish War, Austria—with substantial Imperial support—drove the Ottomans out of Hungary and added it to their realm. At the same time they were able to hold off Louis XIVs attempts to push even deeper into the HRE. Throughout this period, Austria made major efforts to integrate the hereditary lands more closely.

Now granted, Austria was probably one of the weakest of the great powers at the time. Often, they relied on substantial subsidies from wealthier countries (Britain, the Dutch, the Pope) to carry on offensive warfare, but honestly nearly every state in early modern Europe struggled with money.

Hope this helps! Feel free to ask any follow ups.

I mean it depends on how you look at it. In the formal sense of a Great Power there were originally four plus France. Britain, Russia, Austria and Prussia. Basically the main powers of the 6th coalition that stopped Napoleon. These were formalized at the congress of Vienna in 1814. In a formal sense, these are the five great powers at the time the term "Great Power" started to see regular use.

Over time the number of great powers would shift. Italy was considered a great power after its unificatoin and restoration. Prussia was replaced by the unified Germany, the Austrian Empire became the Austro-Hungarian Empire etc. Powers would also wax and wane. As you mention, Austria did not keep up. Generally speaking Austria and the Ottoman Empire (technically not a great power at the Congress of Vienna) stagnated whereas Germany, France and Britain did not. Colonial Empires and rapid industrialization played an important part in this. With time Imperial Japan, Italy and the United States also rose to the status of Great Powers.

One hundred years after the Congress of Vienna there were effectively eight great powers where there had once been 4+1.

That said, there is no strict definition of what constitues a great power other than the willingness and ability to act as a great power. That is, of exert power on a global scale. Various attempts have been made to establish spatial, power and status dimensions for what a great power is but none is, at least to my mind, fully satisfactory.

In 1814 Austria was absolutely a great power. Their contribution to the continental balance of power over the previous century was undeniable and it was at the time the third most populous and third largest in terms of area. However, Austria was also a great power in terms of status long after its power status had started fading. I think George Modelski explained it pretty beautifully with "The status of Great power is sometimes confused with the condition of being powerful. The office, as it is known, did in fact evolve from the role played by the great military states in earlier periods... But the Great power system institutionalizes the position of the powerful state in a web of rights and obligations."

In very basic terms, Austria remained a great power because it had historically been a great power.