The Teutonic Knights were the "German" crusading order. In Germany they are simply known as the German Order. The Templars and Hospitallers were made up of French and English speakers so the German Knights in the Holy Land came together to form the German Order during the 3rd crusade (Crusading Orders provided most of their own hospital care, and the German crusaders needed hospital care for their language speakers, so the Order of Brothers of the German Hospital of Saint Mary in Jerusalem (Teutonic Knights) was founded.)
After the end of the 3rd crusade the Teutonic Order had far less possessions in the holy land than the Templars and Hospitallers, while at the same time there were pagans in need of conversion much closer to Germany in modern day Poland. The Order relocated to Transylvania for a short time and then to Poland. They gained a strong foothold in Poland with support from the Holy Roman Empire who was very close by, and it was much more economical for German Knights to gain salvation by crusading in Poland, a mere week or two travel from most parts of Germany, where as the Holy Land and Spain were much much further away.
They ceased to exist after a bankrupting military defeat at Tannenberg in 1410. Tannenberg began the Order's decline for multiple reasons. The Order was bankrupted by ransoming the knights captured at Tannenberg, and their Pagan enemies had recently converted to Catholicism making the Teutonic Knights military actions illegal in the eyes of the Pope. Following the beginning of protestantism by Martin Luther, who acted dangerously close to the Order's borders, the Order was secularized in 1525, eviscerating their power.
They still in exist today, and their hospitals can be found in nearly every mid-sized German town. The only time the order was ever completely gone, was when the Nazis dissolved the Order in 1938-1939.
TL;DR The Germans needed solidarity in the Holy Land, and crusades stop working when your enemy converts to Christianity.
That's the short and sweet version folks You can find all of this info in The Teutonic Knights: A Military History by William Urban.