Austro-Hungarian Compromise is an 
Austrian decision, and the only way to form 
Austria-Hungary. It can be seen in the decision tab if Austria owns at least 1 province with Hungarian core and it is any form of monarchy (absolute monarchy, Prussian Constitionalism or HM's Government). It is possible to take if both 
Italy and 
Germany exists, the average militancy among Hungarians is below 5 and Nationalism and Imperialism has been invented.
The 1867 Compromise established a dual monarchy between the Austria and 
Hungary. The compromise was a countermeasure against the Italian unification and German unification. It gave Hungary partial freedom and all Habsburg lands were reorganized.
Austria-Hungary will be formed to counteract against Germany and Italy. Austria-Hungary has Hungarian as an accepted culture, this equates to more revenue, administration, and manpower. Austria will lose cores on all provinces, which does not have South German as their main culture, which is most of the empire. All Hungarian POPs will lose 2 militancy, while all non-Hungarian and non-south German will gain 2.
The decision cannot be taken twice, so if Austria-Hungary ceases to exist it is gone forever.
Ingame description
The Nobles of the Hungarian Estates have, for some time, been pushing for increased independence for the Hungarian people and the Crown of St. Stephen. A proposal for compromise has now been drafted, under which the Kingdom of Hungary and the Empire of Austria will amalgamate into a new political entity: The Austro-Hungarian dual monarchy. Gott erhalte, Gott beschütze!
See also
http://en.wikipedia.org/wiki/Austro-Hungarian_Compromise_of_1867