"Barış has a dog."
Translation:Barış'ın bir köpeği var.
"Köpek var" is "There is a dog." The "Barış" is just kind of hanging there in the sentence.
"Barış has a dog" gets literally translated as "Barış's dog exists". So for "Barış's dog", you have to make Barış genitive (Barış + -ın = Barış'ın) and add a possessive "his" suffix to the dog (köpek + -i = köpeği). So you have Barış'ın (bir) köpeği var.
For example, the third-person singular possessive suffix -İ (= -i, -ı, -u, -ü) starts with a vowel, as does the dative case suffix -A (= -a, -e).
But not, for example, the plural suffix -lAr (= -lar, -ler) or the locative case suffix -DA (= -da, -de, -ta, -te), which both start with a consonant.
I think that it's just a matter of grammatical agreement. Using either of these suffixes alone would probably get across the meeting, but the resulting sentence would be agrammatical. (Hopefully somebody who knows more Turkish than I do can confirm this.)
Similarly, in English, if you were to say either ‘There are dog.’ or ‘There is dogs.’, then that clearly indicates your meaning, but it's not grammatically correct; you must say ‘There are dogs.’. The noun and verb must agree in number.
In fact "ta" suffix in "Barış'ta bir köpek var." is not a possessive suffix. It is the locative suffix "-de,da,te,ta". So the literal translation would be "There is a dog at Barış." With places it would maybe make more sense for a non-native: "Evde bir köpek var."="There is a dog at home."
The meaning is though slightly different: when you say "Barış'ta bir köpek var.", the dog does not need to be owned by Barış. Maybe he is looking for someone to adopt it. On the other hand "Barış'ın bir köpeği var." always means that he owns a dog.
I thought the third person possessed ends with an (si) like kedisi, why does the possessed end with just an i here, kopeği?
-i -ı -ü -u is the normal ending.
But because Turkish doesn't like two vowels coming in a row, if you add that ending to a word that ends in a vowel, you need a buffer consonant, which is -s- in this case.
So Barışı, köpeği, evi, hayvanı, … but kedisi, elması, köprüsü etc.