"East Prussia" used to be known as just Prussia. This was the region in which the Prussian people (a group of Slavs) dwelled. As this area was subsequently conquered by the Teutonic Order, ethnic Germans moved into this area and became the predominant group.

After the defeat of the Teutonic Order by Poland, their territories were split into two parts: Royal Prussia (under the direct authority of the Polish monarch) and Ducal Prussia, a duchy under the Kingdom of Poland. This remained the case until it was inherited by the rulers of Brandenburg, whom after a series of wars and treaties eventually were able to get themselves recognized as "King in Prussia," which started the rise of Prussia as a separate entity from the Holy Roman Empire. The Kingdom of Prussia grew in power while Poland declined, and during the Partitions of Poland Prussia annexed Royal Prussia, which henceforth became known as "West Prussia."

After this, Prussia eventually unified Germany, under the federal German Empire. After Germany's defeat in WWI, significant amounts of German territory were handed over to the revived Poland. This included West Prussia. East Prussia was still German, however, and remained so until the end of World War II, where by part of it was given to Poland and the rest remained a Russian enclave to this day.

I believe you are referring to the semi-autonomous period of Danzig between WWI and WWII. Danzig was heavily German: however, to give Poland access to the sea, the League of Nations gave Poland control over Danzig, while giving the city itself autonomy over certain affairs, while still remaining under Polish authority.