The letter ö is used in
Estonian,
Finnish,
German,
Hungarian,
Icelandic,
Swedish, and
Turkish, all with essentially the same sound, a
rounded front vowel, something like the English vowel of
bird, colonel (but without an R sound).
In the Germanic languages (German, Icelandic, Swedish) it generally arises from the umlaut mutation of an O that was once followed by I or J, as in German Vogel 'bird', plural Vögel: pluralization is one common use. The same sound existed for the same reason in early Old English but turned into E, which is why we have pairs such as goose ~ geese and long ~ length.
In Icelandic use of the letter is modern; it replaces the Old Norse letter Ø, pronounced in the same way, but has also subsumed a different letter, O with a hook under it, pronounced like the O in (British) English hot, which arose as a U-mutation of A. For example, the vowel in taka 'to take' changes to Ö in vidh tökum 'we take'. [That dh is meant to be the edh or crossed-curly-d ð (ð), which however is not a friend of browsers.] The U-mutation is still a living feature of Icelandic: the (nominative) drink Fanta becomes Föntu in other cases.
Another origin is borrowing: German Frisör 'hairdresser' is from French friseur, where EU has the same sound.
In Finnish Ö is an alternant in vowel harmony with O: for example the interrogative suffix is -ko or -kö depending on the preceding vowel.
Hungarian has a peculiar letter that I can't write here *cough* Netscape. 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 ő and makes this: ő
In Greek, English, and Dutch it may occasionally be seen in the role of diaeresis, that is separating two vowels that would usually be read as a digraph. The classic example is coöperate, which of course you never see any more written like that.
A kind of Nodeshell Challenge, only this was triggered by the fourth and last line of a now-deleted write-up, which said "(What more can be said about ö?)".