It does not just apply to the
intrinsic structure of a word, but usually to the
suffixes added to it. (Languages that have vowel harmony typically use suffixes a lot, and it might have developed from a strong initial
stress, as in Finnish.) So with
back vowels Finnish goes like this
- talo 'house'
- talossa 'in (the) house'
- talonsa ' her/his house'
- talossansa 'in her/his house'
- talossansako? 'in her/his house?'
whereas with
front vowels you get
- kylä 'village'
- kylässä 'in (the) village'
- kylänsä ' her/his village'
- kylässänsä 'in her/his village'
- kylässänsäkö? 'in her/his village?'
Vowel harmony is common in many North Asian languages of various families; in fact it is used as evidence that they may be genetically related in a hypothetical Ural-Altaic family. On the other hand, more detailed examination suggests that some of the languages may have borrowed the system from others they were in contact with.
It occurs in Finnish and in some related Finno-Ugrian languages, such as Hungarian, Mordvin, and Mari. However it does not occur in Estonian or Sami (Lappish), which historically have never been in contact with Turkic languages.
Hungarian has also developed a threefold harmony, in which the suffix meaning 'to' occurs as -hoz after back vowels, -hez after front unrounded vowels, and -höz after front rounded vowels.
Vowel harmony occurs in Mongolian and related languages, and in Turkish and related Turkic languages: these two groups may be connected in the Altaic family. The other Altaic group, Manchu-Tungus, has it less strongly. It also occurred in Old Japanese, which some linguists believe may be related to Altaic, but has ceased to apply in modern Japanese.
Turkish has a few suffixes that don't change at all, but most suffixes use one of two vowel-harmony systems. The first just distinguishes front from back vowels, using e after front vowels and a after back vowels. So for example with -de/-da 'in, at', they say Türkiyede 'in Turkey' but Ankarada 'in Ankara'.
The other system is fourfold, distinguishing rounded versus unrounded as well as front versus back: suffixes contain front rounded ü or front unrounded i or back rounded u or back unrounded I (an undotted i, which is a different letter from dotted i). Thus
- göz 'eye', gözün 'of the eye'
- el 'hand', elin 'of the hand'
- yol 'road', yolun 'of the road'
- adam 'man', adamIn 'of the man'
I am not sure how widespread it is. I did once see an example in one of the Australian Aboriginal languages, using only the three vowels a i u. A number of African languages have systems of vowel harmony based on plus or minus advanced tongue root, a phonetic feature that's only been understood for about thirty years.