This is a huge question and not one that lends itself to a short, pat answer. For the purposes of this answer, I will be covering the period ca. 1867-1918.

Political

After the Ausgleich of 1867, there were two major administrative divisions within the empire: Cisleithania (the Kingdom of Austria) and Transleithania (the Kingdom of Hungary). The immediate factor that precipitated the Ausgleich was the Austrian defeat in the Austro-Prussian War, but it built upon long-standing political arrangements between the Hapsburg dynasty and Hungarian elites. Moderate Hungarian nationalists realized that compromise with the Hapsburgs, especially in their moment of weakness, would lead to the erection of a Hungarian national state. Each kingdom had its own separate laws, parliament, and army. The Hapsburg monarch would be the head of both governments and oversaw a joint state apparatus that would coordinate the activities of both kingdoms, such as a common military high command. In practice, the system evolved so that wherein the Minister-President of Cisleithania acted as the prime minister and the Hungarian acted as foreign minister (although there were exceptions in the latter case).

The two governments were quite distinctive from each other. The period of 1870-90s is indicative at how far apart both kingdoms could be governed. In Hungary, the state fell under the influence of Kálmán Tisza, a prominent nationalist and a liberal Prime Minister. He pushed for a greater liberalization of the economy and a more thorough Magyarization of Transleithania. His tenure of office was characterized by increased centralization. This was a strong contrast to Tisza's Austrian contemporary Eduard Taaffe. Taaffe crafted a political coalition termed the Iron Ring consisting of Czech nationalists, German and Polish aristocrats, and Catholic ultramontanes. Although initially a liberal, Taaffe moved towards the right and locked German liberals out of Austria's political system with his coalition. Thus Taaffe's conservative ministry excluded German nationalists (many of whom were also liberals), encouraged Czech nationalism, and made alliances with the Polish aristocracy. Meanwhile the Kingdom of Hungary was embracing a more liberal stance and adapting a program to render the state more ethnically homogenous.

This schizophrenic government limped along until the WWI. In general, Hungary became more politically unified in its Magyar core, while causing ethnic tensions at its frontiers. The Austrian Kingdom was becoming more fragmented politically and socially. The important glue that held this political arrangement together was the Hapsburgs, who still enjoyed a degree of political popularity and Franz Joseph I whose longevity imparted to the monarchy an ancient regime luster. The death of Franz Joseph I and the incompetence of the Hapsburg government at managing the war, especially the food supplies, were fatal blows to the ability of the Hapsburgs to act as a unifying symbol.

Culture

This is something of a tricky question. Older historiography exemplified by Carl Schorske emphasized that the political and social fragmentation of Cisleithania contributed to a highly creative period of art during the fin de siecle known as the Viennese Secession. Conversely, cultural historians of Hungary have asserted that Magyarization produced a national style exemplified by figures such as Bartok and Liszt. Newer historians have reversed this picture somewhat, with Secession's elements creeping into Hungarian arts scene and Austria being portrayed as not being a society dancing on the edge of a volcano.

In the immediate decades prior to the war there was a certain popular culture associated around the Hapsburgs. Franz Joseph I's late wife Elisabeth emerged as something of a folk hero throughout the empire and remains a very popular historical figure in Austria today. This imperial kitsch speaks to some sort of trans-national popular culture had emerged within the Empire, even if it was on very shallow foundations.

Social/Economy

Older historiography emphasized the nationalist fragmentation of the Empire almost to the point that the dissolution of the empire becomes a teleological given. This assertion that nationalism is the poison of empires has been qualified by a generation of post-nationalist scholars like Jeremy King and Tara Zahra that assert that national identity was highly fluid and immediate factors precipitated the empire's breakup, not ethnic hatreds. The truth is somewhere in the middle, nationalism ate away at the foundations of the empire, but external events acted as catalysts.

The collective indices for empire's economy grew relatively well during the period of the dual monarchy. One of the reasons for this was the fact that one of the few tools supranational loyalty the Hapsburgs possessed was their commitment to the rule of law. Areas such as Galicia enjoyed an oil boom and Austrian and Bohemian manufacturing grew. Economic growth was regionally uneven with some areas not sharing in this general prosperity. In general, Austria-Hungary had a East-West division in its economy, with the latter economically vibrant and the eastern half of the empire stagnant.

Sources

Cole, Laurence, and Daniel L. Unowsky, eds. The Limits of Loyalty. New York: Berghahn Books, 2007.

Coen, Deborah R. Vienna in the Age of Uncertainty: Science, Liberalism, and Private Life. Chicago: University of Chicago Press, 2007.

Cohen, Gary B. The Politics of Ethnic Survival: Germans in Prague, 1861-1914. Princeton: Princeton University Press, 1981.

Dabrowski, Patrice M. Commemorations and the Shaping of Modern Poland. Bloomington: Indiana University Press, 2004.

Deák, István. Beyond Nationalism: A Social and Political History of the Habsburg Officer Corps, 1848-1918. New York: Oxford University Press, 1990.

Frank, Alison Fleig. Oil Empire: Visions of Prosperity in Austrian Galicia. Cambridge: Harvard University Press, 2005.

Judson, Pieter M. Guardians of the Nation: Activists on the Language Frontiers of Imperial Austria. Cambridge, Mass: Harvard University Press, 2006.

King, Jeremy. Budweisers into Czechs and Germans: A Local History of Bohemian Politics, 1848-1948. Princeton: Princeton University Press, 2002.

Prokopovych, Markian. Habsburg Lemberg: Architecture, Public Space, and Politics in the Galician Capital. West Lafayette: Purdue University Press, 2009.

Schorske, Carl E. Fin-De-Siècle Vienna. New York: Vintage Books, 1981.

Stauter-Halsted, Keely. The Nation in the Village: The Genesis of Peasant National Identity in Austrian Poland, 1848-1914. Ithaca: Cornell University Press, 2001.

Unowsky, Daniel L. The Pomp and Politics of Patriotism: Imperial Celebrations in Habsburg Austria, 1848-1916. West Lafayette: Purdue University Press, 2005.

Zahra, Tara. Kidnapped Souls: National Indifference and the Battle for Children in the Bohemian Lands, 1900-1948. Ithaca: Cornell University Press, 2008.

Wingfield, Nancy M. Flag Wars and Stone Saints: How the Bohemian Lands Became Czech. Cambridge: Harvard University Press, 2007.

Wolff, Larry. The Idea of Galicia: History and Fantasy in Habsburg Political Culture. Stanford: Stanford University Press, 2010.