People from Wales are called the Welsh; one person is a Welshman or Welshwoman, or simply a Welsh person. Merriam-Webster defines “Welsh” as “the natives or inhabitants of Wales,” and the same word names their Celtic language. In the Welsh language itself, the people are the Cymry (singular Cymro for a man, Cymraes for a woman), and the country is Cymru.
The English name has an outside origin. Merriam-Webster traces “Welsh” to the Old English wealh, a word meaning “foreigner” or “Briton” that the Anglo-Saxons applied to the Celtic-speaking Britons. Today Wales is a nation within the United Kingdom, and Welsh people are also British citizens, so both “Welsh” and “British” are correct.
Source: Merriam-Webster, “Welsh”
Image: Flag of Wales, public domain, via Wikimedia Commons.