Turkish is a language of the Ural-Altaic family. It's quite logical, with few exceptional rules and no genders, but its agglutinative structure is so different from Indo-European languages that speakers of those languages may find its grammar a challenge to learn at first.

(Agglutinative means that words and sentences are made by adding suffixes to a root-word.)

Suffixes
A Turkish word starts with a short root (such as git-, 'go'). One or more suffixes are added to modify the root (gitti, 's/he went'). English uses only a few suffixes, such as -'s for possessive, -s or -es for plural, but Turkish has dozens of suffixes. You can make whole sentences in Turkish out of one little word root and a lot of suffixes.

Vowel Harmony
Suffixes are formed according to Turkish vowel harmony, rules whereby most vowel sounds in a word are made either in the front of the mouth or the back, but not both. Details.

Buffer Letters
Suffixes are sometimes preceded by a 'buffer letter' such as 'y' or 'n' for smooth pronunciation.

Stress is usually on the last syllable of a word.

NOUN SUFFIXES

Plural: -lar, -ler
Bankalar, banks
Oteller, hotels

To, Toward: -a, -e (or -ya, -ye)
Bankaya, to the bank
Otele, to the hotel

From: -dan, -den
Bankadan, from the bank
Otelden, from the hotel

Possessive: -ın, -in, -nın, or -nin
Bankanın, the bank's
Otelin, the hotel's

With: -lı, -li, -lu, -lü
Et, meat; etli, with meat
Süt, milk; sütlü, with milk

Without: -sız,-siz,-suz, -süz
Et, meat; etsiz, without meat, meatless
Süt, milk; sütsüz, without milk

You may see -ı, -i, -u or -ü, -sı, -si, -su or -sü added to any noun. An ev is a house; but the ev that Ahmet lives in is Ahmet'in evi.

VERB SUFFIXES

Infinitive: -mak, -mek
Almak, to take or buy
Gitmek, to go

Simple present: -ar, -er, -ır, -ir, -ur, -ür
Alır, he/she/it takes or buys
Gider, he/she/it goes

Future: -acak, -ecek, -acağ-, -eceğ-
Alacak, he/she/it will take, buy
Gidecek, he/she/it will go

Simple past: -dı, -di, -du, -dü
Aldı, he/she/it took, bought
Gitti, he/she/it went

Continuous: -ıyor-, -iyor- (like English '-ing')
Alıyor, he/she/it is taking, buying
Gidiyor, he/she/it is going

Question: -mı, -mi, -mu, -mü
Alıyor mu? Is he/she/it taking (it)?
Gidecek mi? Will he/she/it go?

First Person Singular (I):-ım, -im, -um, -üm
Alırım, I take

Second Person Singular (you-informal): -sın, -sin, -sun, -sün
Alırsın, You take

Third Person Singular (he/she/it): (no suffix)
Alır, he/she/it takes

First Person Plural (we): -ız, -iz, -uz,-üz
Alırız, we take

Second Person Plural (you-formal): -sınız, -siniz, -sunuz,-sünüz
Alırsınız, You (plural) take; or You (singular-formal) take

Third Person Plural (they): -lar, -ler
Alırlar, They take.

WORD ORDER
Nouns and adjectives usually come first, followed by the verb. The subject of the sentence is often the final suffix (unless the sentence is a question):

Ankara'ya gidecegim, I'm going to Ankara.

Kitap almak istiyorum, I want to buy (take) a book (literally 'Book to buy want I')