The Teutonic Knights, or Order of the Hospital of St. Mary of the Germans of Jerusalem (!), were originally founded in 1189 or 1190 at Acre during the Third Crusade. Like the Knights Hospitaler and Templars, they were a crusading order dedicated to keeping Jerusalem under Christian control and aiding Christian pilgrims. Although headquartered in the Outremer (Christian-controlled Palestine), they received large land grants in Europe as well. The largest came in 1225 when the (Polish) Duke Conrad of Masovia turned Prussia over to the Teutonic Knights, after the failure of of another military order to bring the non-Christian inhabitants under Christian rule.
In 1229, the order began conquering their newly-assigned territory, over the objections of Christian missionaries who believed the knights were not particularly interested in actually converting the locals. The Teutonic Knights soon turned this Prussian territory into virtually an independent kingdom by signing it over to Pope Gregory IX in 1234, who granted it back to the order as a papal fief. To be answerable only to the pope was, effectively, to be completely independent. In addition to subjugating the non-Christian inhabitants, the Teutonic knights established a system of colonization and town-building. In 1245, the order obtained from Innocent IV a plenary indulgence (full remission of any penance for sins) for anyone crusading in Prussia. The Teutonic Knights gradually gained control of Prussia (and eventually Livonia, an area now in Latvia and Estonia) by building castles, establishing towns peopled by German colonists, and recruiting Dominican missionary friars to convert the native Prussians. This process was definitely not without setbacks, especially as the Teutonic Knights began encroaching on the relatively strong non-Christian Lithuanian kingdom.
Before the Teutonic Knights had even finished their conquest of Prussia, however, the Mongols began invading Poland in 1241. A crusade against the Mongols was declared by Pope Gregory IX that same year, and the Teutonic Knights were essentially placed in charge of defending Poland from the Mongols. In the meantime, the order was facing the first of two serious revolts in Prussia, which was not fully subdued until 1283. The Teutonic Knights reduced the surviving Prussians to serfdom. Toward the close of the 13th century, the military orders in general (most famously of course the Templars) came in for a great deal of criticism because of the failed crusades in Palestine. Although the holy land was no longer a very significant field of operations for the Teutonic Knights, the clergy in their own territories in Prussia and Livonia criticized them vocally for corruption, impeding missionary work, brutal treatment of the people and clergy, and even plundering churches. In 1310 the pope even placed the order under investigation. In the meantime, the Teutonic Knights were often fighting on two fronts against both the Lithuanians and the Mongols. They were also in nearly constant political conflict with the neighboring Kingdom of Poland.
In response to being assailed from so many different directions, the order through most of the fourteenth century concentrated all its efforts on consolidating and expanding its Baltic fiefdom. The Teutonic Knights turned crusading (mainly against Lithuania) into virtually an aristocratic summer holiday in expeditions called "reysen." Nobles and their retainers from all over Europe came to Prussia to participate.
The order's fortunes took an abrupt turn for the worse in 1386 when the Lithuanian Grand Duke Jagiello was baptized Christian and, by marrying Queen Jadwiga of Poland, became King of Poland. He turned to reconquering Polish territory that the order had taken for themselves. After a series of crippling defeats, the Teutonic Knights signed a 1411 treaty with Poland that did allow them to keep most of their territory, but their strength was broken. Their own subjects grew steadily more unhappy with the heavy-handed rule of the Teutonic Knights. Many Prussian towns formed an independent league, the Prussian Confederation, placing themselves under Polish protection. Between 1454 and 1466, Poland supported the Prussian Confederation in a rebellion against the Teutonic Knights. This ended in defeat for the Teutonic Knights and loss of almost all their remaining Prussian territory, which was now held as a fief from Poland rather than the popes. It was finally the Reformation that ended the order, as the last grand masters of the Prussian and Livonian branches converted to Lutheranism in the sixteenth century and became dukes.
Source: Jonathan Riley-Smith, The Crusades: A Short History
TLDR: The Teutonic Knights were largely responsible for the presence of a significant ethnically and linguistically German population on the Baltic coast, which would have important ramifications even into the twentieth century. They also played a role, if a complicated and not necessarily positive one, in integrating the Baltic coast into Christian Europe. The did play some role in defending Europe against the Mongols.