The word Welsh can refer to the Celtic language of Wales, called Cymric in that language, or it can be an adjective referring to items related to “Wales or its people, language or culture” (AHD s.v. Welsh.
Etymologically, the word Welsh entered Modern English via the Middle English Walische, derived from Old English Wælisc, from Old English Wealh, “foreigner.” The plural form of wealh, wealas, gave us the Modern English word Wales.
There’s a certain irony that the Germanic-speaking invaders refer to the previous inhabitants of Britain, the Celtic speaking ancestors of modern Welsh, as “foreigners,” but to the English, the people “over there,” across the border in Wales, were “foreign,” if not “enemies,” and in some contexts, the word Wælisc does in fact seem to be a synonym for “enemies.”
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As the note in the AHD entry for Wales mentions, Welsh derives from the I.E. root walh-, as does the OE word for walnut; walhhnutu. This word, is a surviving in a single c. 1050 ms. is a compound of wal + nut and refers to the nut of the common walnut tree Juglans regia Juglans regis – Plant Finder . Old English Walhhnutu became Middle English walnotte and Modern English walnut. At the literal level, wal + nut means “foreign nut,” an apropos name for a tree brought to Britain by the Romans, and native to Eastern Europe and Asian minor.