The letter ü is used in
Estonian,
German,
Hungarian, and
Turkish, all with essentially the same sound, a front
rounded high vowel, not occurring in standard English, but something (though not exactly) like the one some younger speakers use in
good*. Some languages with both Ö and Ü sounds use Y instead of Ü: Finnish for one; and French writes it as plain U. The
IPA phonetic symbol is
[y]. To make it, pronounce
[i] as in
machine with rounded lips. The HTML codes for capital and lower-case are Ü and ü
In German it generally arises from the umlaut mutation of a U that was once followed by I or J, as in Buch 'book', plural Bücher: pluralization is one common use.
Hungarian has a peculiar letter that won't show up in all browsers. An acute accent indicates length: so Á is long A, Ó is long O. Hungarian has short vowels Ö and Ü; to make these long replace the two dots with two acutes. In Unicode it's ű and makes this: ű
In Spanish, as noded above, it indicates the U is pronounced, not silent. In French there are rare instances where a similar treatment is required: the feminine of aigu 'acute' would, if simply written aigue, look like a monosyllable, so a diaeresis is added. However, it appears to be a debatable point whether to write aigüe or aiguë. To me the former is more logical because it is the U that is being marked as non-silent.
Both umlaut and diaeresis are processes, one a fronting of vowels in the Germanic languages, the other a separation of two vowels that would otherwise form a diphthong. The accent itself has no separate name: you can call it either. It doesn't matter.
It is also used in the pinyin transcription of Chinese, with the usual Ü sound, but only after some consonants, such as N, where both nu and nü are possible. After Q, X, and Y, a plain U is not possible, so these are written qu, xu, yu but pronounced with Ü.
* A discursion on the sound in good. A fronted U, phonetic symbol [u], has long been common in Scotland: witness the spelling guid. This isn't Ü, it's about midway between U and Ü. Many young people in England now also have a fronted vowel like this, but it's also unrounded, like Japanese U. It seems to be particularly forward in common words like good and cool. I recently noticed that a similar fronting occurs in some American speech.