The Austro-Hungarian Monarchy was a unique entity, where Austria and Hungary were joined into one state: the emperor of Austria was also the king of Hungary. Foreign policy, defense, and financial matters were handled together, but in other matters the two countries enjoyed sovereignty. (At the same time, the kingdom of Croatia-Slavonia also enjoyed some autonomy, but it belonged under the Hungarian crown.)
The 1867 agreement that created Austria-Hungary put an end to the hostility between the two nations that only got worse after the 1848-49 revolution of Hungary, which was brutally crushed and ended in almost 20 years of further oppression. The new form of government, however, allowed for an era of relative autonomy and prosperity in Hungary. Pieces of art and durable commodities (e.g. furniture) created in this period are still referred to as made “during the peace”.
Austria-Hungary was a central power in Europe. Still, its existence was cut short by World War I, at the end of which the Monarchy was split. Various territories of Hungary were attached to newly created neighboring countries, most of which you can still see on up-to-date maps.