Königsberg (perhaps most famous as the inspiration for a namesake math problem involving the city's bridges) was part of the state of Prussia by the mid-1500s and remained part of it up through the unification of Germany under Otto von Bismarck in 1871. Following Germany's defeat in World War I, however, the Germans were forced to cede West Prussia to Poland, in particular the "Polish Corridor," a strip of land that gave Poland access to the Baltic Sea near the port of Danzig (which was designated a "free city" belonging neither to Germany nor Poland). The Polish Corridor cut East Prussia off from the remainder of Germany, rendering it - and its key city, Königsberg - an exclave.
The German Empire at the time of the first World War. Königsberg is located just northeast of Danzig across that small inlet of the Baltic Sea known as the Gulf of Danzig.
World War II-era map of the "collapse" of Poland, showing Germany's reoccupation of Polish territory around the former East Prussian exclave.
The Conference examined a proposal by the Soviet Government to the effect that pending the final determination of territorial questions at the peace settlement, the section of the western frontier of the Union of Soviet Socialist Republics which is adjacent to the Baltic Sea should pass from a point on the eastern shore of the Bay of Danzig to the east, north of Braunsberg-Goldap, to the meeting point of the frontiers of Lithuania, the Polish Republic and East Prussia.
The Conference has agreed in principle to the proposal of the Soviet Government concerning the ultimate transfer to the Soviet Union of the City of Koenigsberg and the area adjacent to it as described above subject to expert examination of the actual frontier.
Just why the Soviets were so hot for Königsberg - which they renamed Kaliningrad in the late 1940s - is a bit vague; Stalin declared it important to Soviet interests, so presumably they just wanted another major Baltic port. This was no problem until the breakup of the Soviet Union in 1991; previously, one could drive through Soviet territory uninterrupted from Moscow to Kaliningrad. But with Lithuania, Latvia and Belarus becoming independent, suddenly Kaliningrad was two countries away by ground in any direction! At a distance of more than 200 miles from the nearest Russian frontier, the Kaliningrad Oblast today is one of the world's farther-flung exclaves.
Google map of Kaliningrad today, two countries away from the nation to which it belongs.
Non-linked references:
Map of Germany and Europe 1914 from http://www.fsmitha.com/h2/map01eu.htm
"Polish Collapse" map from http://history.sandiego.edu/cdr2/WW2Pics/51867big.jpg