Galli, Galatai, Welsh, and Gael
Latin Galli might be from an originally Celtic ethnic or tribal name, perhaps borrowed into Latin during the early 5th century BC Celtic expansions into Italy. Its root may be the Common Celtic *galno-, meaning "power" or "strength". The Greek Γαλάται Galatai (cf. Galatia in Anatolia) seems to be based on the same root, borrowed directly from the same hypothetical Celtic source which gave us Galli (the suffix -atai is simply an ethnic name indicator).
Schumacher's account is slightly different: He states that Galli (nominative singular *gallos) is derived from the present stem of the verb which he reconstructs for Proto-Celtic as *gal-nV- (it is not clear what the vowel in the suffix, marked as V, should be reconstructed as), whose meaning he gives as "to be able to, to gain control of", while Galatai comes from the same root and is to be reconstructed as nominative singular *galatis < *gelH-ti-s. He gives the same meaning for both reconstructs, namely "Machthaber", i. e. "potentate, ruler (or even warlord)", or alternatively "Plünderer, Räuber", i. e. "raider, looter, pillager, marauder", and points out that both names can be exonyms in order to explain their pejorative meaning. The Proto-Indo-European verbal root in question is reconstructed by Schumacher as *gelH-, whose meaning is given as "Macht bekommen über", i. e., "to acquire power over" in the Lexikon der indogermanischen Verben.
The English Gaul and French: Gaule, Gaulois are unrelated to Latin Gallia and Galli, despite superficial similarity. They are rather derived from the Germanic term walha, "foreigner, Romanized person", an exonym applied by Germanic speakers to Celts, likely via a Latinization of Frankish *Walholant "Gaul", literally "Land of the Foreigners/Romans", making it partially cognate with the names Wales and Wallachia), the usual word for the non-Germanic-speaking peoples (Celtic-speaking and Latin-speaking indiscriminately). The Germanic w is regularly rendered as gu / g in French (cf. guerre = war, garder = ward), and the diphthong au is the regular outcome of al before a following consonant (cf. cheval ~ chevaux). Gaule or Gaulle can hardly be derived from Latin Gallia, since g would become j before a (cf. gamba > jambe), and the diphthong au would be unexplained; the regular outcome of Latin Gallia is Jaille in French which is found in several western placenames.
The French term for "Welsh" is gallois, which is, however, not derived from the Latin Galli, but, like gaulois, borrowed (with suffix substitution) from Germanic *walhiska- "Celtic, Gallo-Roman, Romance" or its Old English descendant wælisc (= Modern English Welsh). The English form "Gaul" (first recorded in the 17th century) and "Gaulish" come from the French "Gaule" and "Gaulois", which translate Latin "Gallia" and "Gallus, -icus" respectively. In Old French, the words "gualeis", "galois", "walois" (Northern French phonetics keeping /w/) had different meanings: Welsh or the Langue d'oïl, etc. On the other hand, the word "Waulle" (Northern French phonetics keeping /w/) is recorded for the first time in the 13th century to translate the Latin word Gallia, while "gaulois" is recorded for the first time in the 15th century, and the scholars use it to translate the Latin words Gallus / Gallicus. The word comes from Proto-Germanic *Walha- (see Gaul: Name).
The English word "Welsh" originates from the word wælisċ, the Anglo-Saxon form of *walhiska-, the reconstructed Proto-Germanic word for "foreign" or "Celt" (South German Welsch(e) "Celtic speaker", "French speaker", "Italian speaker"; Old Norse "valskr", pl. "valir" "Gaulish", "French"), that is supposed to be derived of the name of the "Volcae", a Celtic tribe who lived first in the South of Germany and emigrated then to Gaul.
The Germanic term may ultimately have a Celtic source: It is possibly the result of a loan of the Celtic tribal name Volcae into pre-Germanic, *wolk- changing according to Grimm's Law to yield proto-Germanic *walh-. The Volcae were one of the Celtic peoples who for two centuries barred the southward expansion of the Germanic tribes (in what is now central Germany) on the line of the Harz mountains and into Saxony and Silesia.
In the Middle Ages, territories with primarily Romance-speaking populations, such as France and Italy, were known in German as Welschland as opposed to Deutschland, and the word is cognate with Vlach and Walloon as well as with the "-wall" in "Cornwall". Other examples are the surnames "Wallace" and "Walsh". During the early Germanic period, the term seems to have been applied to the peasant population of the Roman Empire, most of whom were in the areas immediately settled by the Germanic people.
The term Gael is, despite superficial similarity, also completely unrelated to Galli, see Gaels#Terminology.
Read more about this topic: Names Of The Celts