Yui (結 or 唯) is a Japanese name with multiple kanji possibilities. The character 結 means "to bind" or "to tie" — implying connection and unity. The character 唯 means "only" or "unique." Families choose the kanji that fits the meaning they want.
Yui has been one of the most popular girl names in Japan for the past two decades.
Yui does not currently appear in the US Social Security Administration's top 1,000 girls' names, so we don't publish a US rank or birth count for it. That says nothing about the name's standing elsewhere in the world — only that it sits outside the ranked US data we rely on.
In Pythagorean numerology the letters of Yui reduce to 1, The Leader. This is a traditional interpretive system, not a factual claim about the name.