From Latin.
“Deutsch” and “Teutonic” are cognates, and Teutonicus is the synonym from Latin Germanus.
The old Germanic demonym appears to have been thiudisk (Frankish/Old Franconian) / thiudisc (Old Saxon) / Þēodisc (Old English/Anglo-Saxon) / diutisc (Old High German). This was latinized as Teutonicus (Cimbri et Teutones, anyone). The stem word is the proto-Germanic Þeudā (“people”). This was recorded as the Gothic king Theodoric as Þeudārik (“ruler of the people”).
Due to 2000 years of linguistic evolution, this demonym has evolved into High German Deutsch, Low German düütsch, East Frisian düütsk, West Frisian dútsk, Sölring North Frisian dütsk, Öömran North Frisian tjüsch, Scandinavian tyska...) The Englisn demonym for the Netherlands, Dutch, Dutch duits/diets, is from the same stem, as is Italian tedesco.
Germania is Latin, and means “land of the brethren” (Latin germanus, "siblings with the same parents or father"), which has cognates in Catalan, germà, Spanish, hermano and Portuguese irmão, meaning "brother". The tribes were united with blood oaths, hence “brethren”.
“German” superseded “Almain” (Latin Alemanni, one of the Frankish tribes) in English in the Late Middle Ages. This name still survives in France (Allemagne), Spanish (Alemania) and Portuguese (Alemanha). The Alemanni lived in the territory where both Danuvius (Danube) and Rhenus (Rhine) originate.
We Finns call Germany as Saksa. It is the cognate for Sachsen (Saxony). The Hanseatic merchants which arrived to Finland in the Middle Ages were primarily Saxons.
