Originally one of several monastic orders who fought in the crusades, the Teutonic Order would later conquer its own country, Prussia, which became synonymous with the name of the Order.
In 1227, the Teutonic Order came into the Kulmer Land (East Prussia, at the Vistula), after they had been thrown out of Transsylvania. In 1237, they united with the Livonian Brotherhood of the Sword. The OTL battle of Liegnitz (where many knights fell) never happened, so the Order hadn't to pay a high blood toll; OTOH, the stronger Russia also deterred them from attacking, so there never was an Alexander called Nevsky.
Since 1247, they interfered in the Hessian-Thuringian War of Succession, supporting Sophie of Brabant and her son Heinrich "the Child". 1254, they defeated and killed Count Otto I of Nassau, so his brother Walram II got all of Nassau.
But most of their work was done in pagan Prussia and Lithuania. They were supported in this by Otakar Przemysl, after whom the city of Herzogsberg (at the site of OTL Königsberg) was named in 1255. For example, he helped them personally suppressing a big uprising of the Prussians in 1257. In 1260, the latter were subjugated.
1263, the (converted) Lithuanian leader Mindaugas was murdered by his own people, who became pagan again. The stronger settlement of Germans in Prussia made them feel threatened. Although they probably could've been quite successful if they had hidden in their dense forests, they dared to leave them and attack the Teutonic knights on their own turf. This only lead to their defeat. 1270, the Teutonic Knights decided to subjugate the Lithuanians too, as they had done with the Prussians. For a start, in 1277, Zemgale and Samogitia were conquered by them.
All the time, the headquarter of the Order had still been in Akko, Palestine. Only after the fall of the Crusader States in 1281, they moved it from Akko to Venice, and in 1300 they went to Marienburg at the Vistula.
1289, all of Lithuania was subjugated and since then administered by the Teutonic Knights. Later, it would become the province of Littauen. At the moment, however, the Teutonic Knights were mostly restricted to the valley of Memel / Nyemen river.
1301, Poland started to feel threatened by the Order (which they had once called) and the Przemyslids. Boleslaw V tried to fight them to break free, so a coalition of the Teutonic Order, Bohemia, Moravia and the HRE under king Otto IV of Brandenburg formed against him. In 1308, the war between Germans and Poles didn't end in the latter's favor. In the peace, the Teutonic Order acquired Pommerellen (OTL West Prussia, the Poles also call it Eastern Pomerania), cutting Poland off from the sea.
1315, the Teutonic Order decided to go one step further and invade Russia. The situation was good, since the Russian princedoms were more divided than ever. 1322, the princedom of Polozk was conquered by them. Parts of the population fled. The former princedom was germanized through the centuries, since the Germans still quelled to the East. Polozk itself became later known under the name of Plotzeck.
One generation later, in 1348 Turov-Pinsk was conquered as well, and although the area never was fully germanized, the former capital became the German city of Thurau. The Order became even mightier when they 1360 bought Estonia from Denmark. And 1375, in an alliance with Hungary, the Teutonic Order conquered the princedom of Halicz-Volhyn, which was divided between them. The Teutonic part became object of German immigration and germanization too, and later formed the province of Wolhynien.
1396, the lands of the Teutonic Order were struck by Black Death. Poland had been luckier in this way; thus, king Boleslaw VI attacked them 1404, won the first Teutonic-Polish War; Pommerellen became Polish again. 1426, Boleslaw VI of Poland allied with Birger II Eriksson of Sweden against the Teutonic Order, and 1432 they were victorious again; Poland kept its conquests from the last war and also got Wolhynien, Sweden got Estonia (that's only the northern half of OTL Estonia, though). Only the mediocrity of later Polish kings saved the Order from worse.