The idea is that in any given word (in the case of Finnish and similarly agglutinative languages, this counts words as portions of compounds, Kahvi and Kuppi, as opposed to entire compounds, Kahvikuppi), all the vowels must be in harmony with each other. That is, front vowels and back vowels cannot be in the same word.* Since base words already fit these rules in (keeping the previous example) Finnish, this primarily is an effect on suffixes. Take, for example, the ending -lainen (in English phonetics, lie-nen), meaning "from." For words like Suomi (Swo-me) or Turku (Tour-ku), where the initial syllables have back vowels, the back voweled ending is left unchanged, so we get Suomalainen, "from Finland," and Turkulainen, "from Turku." On the other hand, we have Jyväskylä (Yü**-va***-skü-la***). All front vowels. So, we get Jyväskyläläinen, "from Jyväskylä." Further, for words that have back vowels, but none in the initial syllables, when endings are added back vowels in the ending AND in the non-initial syllables mutate to front vowels. For example, Espoo (es-po); Espööläinen (Es-per-la***ee-nen).
Speaking of Uralic and Altaic languages, two musical things I've been all about lately: Tuvan throat singing (I REALLY want to make some metal with a throat singer) and the new Korpiklaani album Tervaskanto. The first album they've done where most of it is in Finnish. I'm in love.
*In Finnish, there eight vowels. Ä, Ö, and Y are front, A, O, and U are back, and I and E are neutral.
**Ü has no real English equivalent. It's sort of like an "oo" sound if you purse your lips.
***These a's are short, as in "hat".