Note: some of these forms specifically mean "the German language" (like Chinese, Japanese, and Hebrew), while others are all purpose adjectives.
The English word German comes from Latin germanus, germani "German," originally the name of a specific tribe. The name was eventually applied by the Romans to all Germans, and the name Germania was given to the land of the Germans. A possible etymology for this tribal name is "spear men" gari "spear" + mann "man." Other words that are related to the Latin/English term are Modern Hebrew גרמנית germanit, Hindi जरमन jarman, and Indonesian jerman.
German deutsch is from Old High German diutisg and Low German theodisc, meaning "of the people," which comes from an Indo-European root, teuta-, meaning "tribe" or "clan." This became tuath "people" in Old Irish and theod "people, nation in Old English."
The Italian tedesco comes from this root, as do the Swedish word tysk and the English word Dutch, which formerly was applied to all Germans, but gradually came to refer only to those Germanic people who inhabit what is now the Netherlands. In Dutch, German is duits. The medieval Dutch word diets, meaning "Dutch" has been replaced by nederlands, but was more recently used by nationalist-types (in Dutch and in Afrikaans) to refer to the "pan-Netherlandic culture" of the Netherlands, Belgium, and South Africa. East Asian words for German tend to derive from "deutsch" — Japanese ドイツ語 doitsugo, Korean 독어 dogŏ, Chinese 德语 deyu, and Vietnamese Đức. The English word Teuton(ic) is also ultimately derived from this root, and originally referred, in its Latin form, to a specific Germanic tribe from Jutland who migrated into southern Gaul and were defeated by a Roman army in 102 BCE, one of the earliest Roman encounters with an invading Germanic tribe.
Another major group of words for German is represented by allemand (French), alemán (Spanish), alemão (Portuguese), alemany (Catalan), alman (Turkish), Arabic and Farsi الماني almānī. These come from Latin alamanus, which refers to a specific Germanic tribe — the Alemanni. By late Roman times the Alemanni, a splinter group of the Suevi (who gave their name to Swabia), occupied territories on both sides of the Rhine (present day Alsace, Baden, and parts of Switzerland). The high German dialect spoken in this region is known as Alemannic (alemannisch in German). This is a case of a relatively narrow ethnic name being widened to refer to a large group. The name of this group literally means "all men" (alle Männer in modern German)
The words for "German" in Slavic languages — немец niemiets (Russian), niemiec (Polish), němec (Czech), nemac (Croatian), немец nemets (Bulgarian), німец (Ukrainian) — come from the common Slavic root nem, meaning "mute" (немой nemoi in Russian). This fits a common pattern of ethnonyms, by which a people who do not understand the language or cannot speak it well are named "mute" by the speakers of a language or group of more closely related languages. It originally applied to all non-Slavs, but the Germans were the non-Slavic people the Slavs had the most contact with, so the name stuck on them. The Hungarian word for German, német, is adopted from the Slavic root, as is the Romanian nemţesc.
In Finnish, Germany is Saksa after the Saxons.
Sources: André Cherpillod, Dictionnaire étymologique des noms géographiques, Various online dictionaries, Ask A Linguist Archives, thirteenkiller, apoivre, rydel23, lemur_man, pne
Please let me know what I missed and what I screwed up :)
Edit 7 May 2003 19:11. Thanks to all who helped.