Finnish, like all Finno-Uralic languages, has vowel harmony and it affects what vowels go with which words. It also affects the postpositions and endings of words.
In Finnish, there are eight vowels, a, e, i, o, u, y, ä and ö. They are grouped into three groups; front, neutral and back vowels. See the diagram:
The vowels in blue are front vowels (or "hard"), the vowels in green are neutral and the vowels in yellow are back vowels (or "soft").
In Finnish, single words contain only vowels of the same type as the initial, stressed syllable. So, for example, the word, aaltoileva contains only neutral and back vowels, while äidillä contains only front and neutral vowels. With respect to vowel harmony, compound words are separate words, e.g. työmaa, with -ta suffixed työ|maa|ta.
When a word contains only neutral vowels, its suffixes use front vowels. For example, when the postposition -lla or -llä is added to kieli, it becomes kielellä, not kielella.
As a consequence, Finnish speakers often have trouble pronouncing foreign words which do not obey vowel harmony. For example, some Finns may pronounce olympia as olumpia because of the difficulty in producing the sounds of o and y in a row.
Vowel harmony applies to foreign words, too. For example, chat is heard as tsät and consequently the frequentatitive verb "to chat" is tsättäillä (also spelled chattäillä, chattailla, chättäillä, etc.). (Note: Is this really about "vowel harmony" if there's only one vowel in the word "chat"? ch turns into ts because producing the ch sound takes effort, and ts does not.)
i → e
You may have noticed that nimi (name) changes to nimesi (your name), and Suomi (Finland) changes to Suomesta (from Finland). This is because there is an i → e mutation. This mutation is where on some suffixes like -n, -a, -ä, -si changes the last vowel from i to e.
This is a slight verbal mutation that makes this change in the word, that is reflected in this mutation. It affects words that are less than two syllables and it usually is the "old" words like nimi and suomi and it is something that you have to learn by heart because it is not regular. The word taksi (taxi) does not change to taksessa (because it is a new loan word), instead it is taksissa.
The k, p and t mutation
Like the i to e mutation, when you put a suffix after a word, some consonants change when you agglutinate. For example Helsinki changes to Helsingissä after you put a suffix, here -ssä, after it. This is called consonant gradation and it mainly affects the consonants k, p and t. So words like pankki change to pankin. Here is a short, not a complete list:
|initial ending||new ending||result|
|-kk:||-k:||pankki → pankista (bank → from the bank)|
|-pp:||-p:||kauppa → kaupasta (shop → from the shop)|
|-tt:||-t:||tyttö → tytön (girl → girl's)|
|-nk:||-ng:||Helsinki → Helsingistä (Helsinki → from Helsinki)|
|-uku/-yky:||-uvu/-yvy:||puku → puvusta (suit → from the suit)|
|-k:||-:||Turku → Turusta (Turku → from Turku)|
|-t:||-d:||katu → kadulta (street → from the street)|
|-p:||-v:||hupi → huvit (amusement → amusements)|
|-rt:||-rr:||murto → murron (break-in → of the break-in)|
|-ti||-si||löytää → löysi (to find → found)|
Now there is a lot to go through, but take your time and hopefully this will help you to no end. But beware that there is more to go through, as we haven't covered the suffixes yet!