The actual Teutonics were not originally a fighting company, but were instead an offspring from the already founded Hospitallers of the 12th and 13th Century. The need for defense on the hospitals treating those Crusaders who had fallen to the Infidels became necessary, and a Hospital in Germany picked up the slack where the Hospitallers left off.
Originally formed from the German aristocracy around and serving at the hospitale sancte Marie Theutonicorum Jerosolimitanum (the Hospital of St. Mary of the Germans of the Jerusalem, a mouthful, but this is a history lesson), the group of soldiers and knights became known as the der orden des Duschen huses (the order of the German houses, another mouthful), though they went through quite a few incarnations before settling on the Teutonic name.
Though active for some time before, they were not a sanctioned company in the service of the church until Pope Innocent III gave them rights as such just about halfway through the Crusades. Strangely enough, many of their numbers were called from the chruch, with monks fighting alongside knights in defense of the Christians under their care. It was about this time that the Company broke free of the Hospitallers, making with the Templars, one of the three major Christian companies in the Crusades.
Noted for their strict moral code, number of nobility in the ranks, and the unusual belief that only Germans could serve with them, this kept their numbers relatively low compared to the other two, but this left them no less powerful. They were a major force in the spreading of Christianity, pushing as far east as Russia. Also, for their numbers pulled from the Church itself, they were rarely stationed in monastaries, maintaining instead garrisons throughout the land. Their main garrison (who's name has completely slipped my mind) was finally lost and sacked, but not without severe losses on behalf of the sackers.
The end of the Crusades did not end their time, and they continued to spread the word of God throughout the land, maintaining numbers even to this day. Though no longer a 'fighting' company, their deeds of support, salvation, and humanitarion work, the name Teutonics still shines through the ages. Vows of poverty, chastity, chivalry, and honor are still important to the company. Long may they reign.