Short answers, pulled from the story.
Austria-Hungary was founded with the Austro-Hungarian Compromise of 1867 and lasted until 1918, a span of roughly fifty years. It was dissolved after Hungary terminated the union with Austria at the end of World War I, with the armistice signed at Villa Giusti on the 3rd of November 1918.
Austria-Hungary was the second-largest country in Europe by area, surpassed only by Russia. It was also the third-most populous European state, behind Russia and the German Empire, and ranked among the ten most populous countries in the world.
The Compromise of 1867 was the agreement that created the Dual Monarchy by making Austria and Hungary co-equal sovereign states sharing a single monarch. It established three common ministries covering foreign affairs, defence, and the finance needed to fund those two portfolios, while all other government functions remained separate.
Austria-Hungary collapsed because of military defeat, ethnic nationalism, and economic breakdown. Inflation rose from an index of 129 in 1914 to 1,589 in 1918, morale collapsed across the diverse nationalities, and the Allied powers supported independence movements. By late October 1918, Czechoslovakia, Yugoslavia, and other successor states had declared independence, and Hungary formally terminated the union on the 31st of October 1918.
The assassination of Archduke Franz Ferdinand in Sarajevo on the 28th of June 1914 gave Austria-Hungary the pretext it sought to confront Serbia militarily. Austria-Hungary secured German support on the 5th of July 1914 and then issued the July Ultimatum; when Serbia partially refused one of ten demands, Austria-Hungary declared war on the 28th of July 1914, triggering the chain of mobilizations that became World War I.
Austria-Hungary included Germans, Hungarians, Czechs, Poles, Croatians, Slovaks, Slovenes, Romanians, Serbs, Ukrainians, Italians, Jews, and other groups. Emperor Franz Joseph himself spoke German, Hungarian, and Czech fluently, and Croatian, Serbian, Polish, and Italian to some degree, reflecting the empire's multilingual character.