The abbreviation k. u. k. (or k. und k.) stands for for kaiserlich und königlich -- German for Imperial and Royal.
The phrase refers to the so-called "Dual Monarchy" of Austria-Hungary: The Emperor of Austria also reigned, in personal union, as the King of Hungary. All government acts took place in the name of "His Imperial and Royal Majesty": most of the time abbreviated to "k. u. k.". Also, all governmental institutions and offices had a name prefixed with "k. u. k.".
The ubiquity of this phrase in all administrative matters made it a synonym for the Dual Monarchy, often referred to even today as "the k. u. k. monarchy".
In Czech the equivalent abbreviation appears as "C. a K.", in Polish – as "C. i K." or "CK".
Arising from the term "k.u.k." we find the name "Kakania", sometimes used to describe the Empire as a state of mind, bureaucratic and with a very stratified formal society (e.g. in work of Robert Musil). The name "Kakania", by means of a childish German-language pun, also expresses the idea of "shittiness".
Political importance of the "U"
Prior to 1867, the collection of territories under the control of the Habsburg monarch in Vienna used the style of the kaiserlich-königlich (Imperial-Royal) realm, (abbreviated k.k. or k-k) with the hyphen connecting the terms and implying an indifferent or even interchangeable distinction between the various lands. The Habsburg monarch ruled Austria and the Holy Roman Empire as Emperor, and the kingdoms of Bohemia and Hungary as King.
After the crisis and compromise in Hungary resulting in the Ausgleich of 1867, the nationalist Hungarians rigidly insisted upon the und ("and") replacing the hyphen in all usage, in order to indicate the independent nature of their relationship to the other (Austrian) lands.