There are two vowel harmonies in our language and these are the mainest matters. Almost every suffixes are determined according to the vowel harmonies. This matter has been told before too in this site. But like I said in the description, I´m writing this so that the learners get one source more. I shall tell the major/big vowel harmony now.

Before starting to tell major vowel harmony, let me talk about some attributes:

As first, no vowel harmonies are looked in monosyllabic words and compound words.

As second, the suffixes are determined according to the last vowel of the word. 

If it is said that a word is obeying the major vowel harmony, you should understand that word has only hard or only soft vowels. The major vowel harmony concerns all the vowels in the word together.

dergi (magazine) : The vowels of "dergi" are "e" and "i". Both of them are soft vowels. Therefore "dergi" obeys to major vowel harmony.

aralık (1space (distance), 2 December) : "a", "a", "ı". All of them are hard vowels. "aralık" obeys.

pencere (window) : "e","e","e". It obeys.

makine (machine) : "a" is hard but "e" and "i" are soft. "makine" doesn´t obey.

kitap (book) : "i" is soft, "a" is hard. "kitap" doesn´t obey.

sıcak (hot) : because both of "ı" and "a" are hard vowels, "sıcak" is a word that obeys to major vowel harmony.

koç (ram) : "koç" has only one vowel. Therefore no vowel harmony can happen.

Let´s go on giving example by using a few suffixes.

The logic is this:

While deciding about bringing suffix in aspect of the vowel harmonies, look at the last vowel of the word. Bring the suffix according to it. If it is hard, bring the suffix with hard vowel; if it is soft, bring the suffix with soft vowel.

1) Mastar eki (Infinitive suffix) : -mek/-mak

You see all the verbs end with "-mek" or "-mak" if you look up the dictionary. Perhaps you had asked "What is the reason in determining "-mek" or "-mak"?"

Let´s choose a verb randomly: bit- (1end up, finish, 2grow)

bit+mek/mak-->bitmek (1to end up, to finish, 2to grow)

Why?

Because "i" is soft vowel. "a" of "-mak" isn´t soft, it´s hard. So as to keep the major vowel harmony, "-mek" has to come. Thus the word obeys to major vowel harmony because its vowels are completely soft: "i", "e"

ol- (be, happen, become)

ol+mek/mak-->olmak=1to happen, 2to become, 3to be

Öğretmen olmak istiyor. (He wants to become teacher).

Burada olmak istemem. (I don´t want to be here.)

özle- (miss)

özle+mek/mak-->özlemek (to miss)

çok özlemek (to miss very much)

If we brought "-mak", the vowels of the word would be ö, e, a. "ö" and "e" are soft. So as to keep the major vowel harmony, a soft vowel must come. So that they become "ö", "e" and "e", "-mek" must come.

anla- (understand)

anla+mek/mak-->anlamak=to understand

If we brought "-mek", the vowels would be "a", "a" and "e".

Konuyu iyi anlamak (To understand the matter well)

inan- (believe, trust)

inan+mek/mak-->inanmak=to believe in, to trust

The last vowel of the verb is "a". Therefore a hard vowel should have come if it would come.

Sana inanmıyorum. (I don´t believe in you). (direct: I am not believing to you)

oku- (read)

oku+mek/mak-->okumak=1to read, 2to study in a school, to take education in an official school

Üniversitede okuyorum. (I´m studying at the university)

Çok kitap okur. (He reads book very much/Her read many books)

2) Birliktelik eki (Togetherness suffix (with), (by)) : ile, -le/-la, -yle/-yla

bilgisayar (computer)

bilgisayar ile --> bilgisayar+le/la-->bilgisayarla=with the computer, by the computer

a: hard vowel
-la: including hard vowel 

Şu an bilgisayarla meşgulüm (I´m busy with the computer now)

dede (grandfather), dedem(my grandfather)

dedem ile -->dedem+le/la-->dedemle=with my grandfather

Bu konuda dedemle aynı düşüncedeydim. (I had got the same idea on this matter with my grandfather) (direct: I was on the same idea on this matter with my grandfather)

gitar (guitar)

gitar ile -->gitar+le/la-->gitarla=with the guitar, by the guitar

The last vowel of "gitar" is hard. Therefore if a suffix is brought, the word should continue by a hard vowel.

Şarkılara gitarla eşlik etti. (He accompanied the songs by the guitar)

suç (crime)

suç ile--->suç+le/la-->suçla (with the crime)

Hangi suçla yargılanacaklar? (On which crime will they be judged?)

kanıt (evidence)

kanıt ile--->kanıt+le/la--->kanıtla (with the evidence)

Davayı bu kanıtla kazandık. (We won the case by this evidence.)

3) Çoğulluk eki (Plural suffix (-s,-es,-ies)) : -ler/-lar

resim (picture)-->resim+ler/lar-->resimler=pictures

So that the word continues by obeying to major vowel harmony...

Senin bu resimlerini hiç görmemiştik. (We had never seen your these pictures)

din (religion) --> din+ler/lar-->dinler=religions

Çocukluğumda dinlerle çok ilgiliydim. I was so interested in the religions in my childhood. (direct: I was so interested with the religions in my childhood.)

dua (blessing) ---> dua+ler/-lar-->dualar=blessings

dua+lar+la-->dualarla=with the blessings, 

Tören bu dualarla bitti (The ceremony ended up with these blessings)

4) Bulunma hâl eki (Location case suffix (in/at/on)) : -de/da/te/ta

alfabe (alphabet)

alfabe+de/da-->alfabede=in the alphabet

The last letter is soft: "e"

Alfabede yok ama biz telaffuz ediyoruz. It is absent in the alphabet but we pronunciate it.

masa (table)

masa+de/da-->masada=at the table

Tabaklar masada. The plates are on the table.

nere (which place, where)

nere+de/da-->nerede=at which place, where

Çantam nerede? Where is my bag?

alan (1area, 2branch, 3square, 4....5...)

alan+de/da-->alanda

Hangi alanda uzmansın? On which branch are you expert?

Alanda binlerce insan vardı. There were thousands of people at the square.

Now let´s talk about some exceptional suffixes. Why are these exception? Because they don´t take care about the vowel harmony. I´m gonna explain one or two of them.

1) sıfat yapan -ki (the -ki which makes adjective)

balkon (balcony)

masa (table)

masa balkonda (the table is at the balcony)

balkondaki masa (the table that/which is at the balcony)

her şey (everything)

ev (house, home)

her şey evde (everything is at home)

evdeki her şey (everything which is at home)

liman (harbour)

gemi (ship)

gemiler limanda (the ships are at the harbour)

limandaki gemiler (the ships that/which are at the harbour)

yer (1place, 2floor)

halı (carpet)

beyaz (white)

beyaz halı (white carpet)

beyaz halılar yerde (the white carpets are on the floor)

yerdeki beyaz halılar (the white carpets that/which are on the floor)

yerdeki halılar (the carpets that/which are on the floor)

yerdeki halılar beyaz (the carpets that are on the floor are white)

istisnalar:

dün (yesterday) - dün+ki--->dünkü

Dünkü film çok güzeldi. (The film which was yesterday was so nice)

bugün (today) - bugün+ki--->bugünkü

O sınav bugünkü sınavdan zordu. (That exam was more difficult than the exam which is today)

There are also some exceptional words for major vowel harmony. Even if their last vowel is hard, they take suffixes with soft vowels.

A) saat (1the watch, 2hour)

harf (letter)

harp (battle, war)

Kril alfabesinin harfleri göze hoş geliyor. The letters of Cyrillic alphabet are sightly. (direct: The letters of Cyrillic alphabet are coming to the eye nicely)

B) hâl (1case, 2situation, 3position)

ihmal (neglection)

festival (festival)

petrol (petrol)

Kemal (a male name) (see: Mustafa Kemal Atatürk)

final (1the final, 2the final exams in the university)

meşgul (busy)

faul (fault (sport))

alkol (alchohol)

kalp (heart)

rol (role)

moral (morale)

"l" is softening the last vowel which is in front of it if those words are foreign origined. Thus a soft vowel is being needed so that the major vowel harmony continues. 

O petroller çok kaliteli. (Those petrols are so qualified.)

Festivaller düzenlendi. (The festivals have been organized.)

Meşgulüm. (I´m busy)