This is a stereotype word. It does not describe one German person; it points at a picture people carry of Germans in general. The qualities in that picture are order, precision, hard work, and a certain coldness. You meet it most in writing about machines, timetables, sport and work: Teutonic efficiency, Teutonic thoroughness, Teutonic build quality. The writer usually means it as praise, or as half-joking praise. Because it is a stereotype, some readers find it old-fashioned and some find it insulting, and the source marks it both dated and offensive. That is worth knowing before you use it: you may mean a compliment and be heard as a label.

This is the historical use, and it carries no judgement about anyone alive. It points back to tribes who lived in northern Europe long before there was a country called Germany. You meet it in history books, museums and articles about old stories: Teutonic mythology, Teutonic tribes, the Teutonic Knights. Because it is about the distant past, this use is safe in a way the character use is not. If you are writing about modern Germany, this is not the word you want.