ki - that, that is

1. That, That is
We do not use it like the "that" in English always, indeed we can, but it is just not that common.

example;
He said that he will go.
O gideceğini söyledi. is the form we use most.
But it is also possible to say; O dedi ki yarın gidecekmiş.
(not to make you confused, demek : to say, söylemek : to tell)
and for the first one;
masadaki kitap : the book (which/that is) on the table
filmdeki adam : the man (who/that is) in the movie
if someone ask which book? then the answer is masadaki
and the answer to which man? is filmdeki likewise.

2. To Put Emphasis
When we take away -ki, the meaning does not change much, it adds just some emphasis.
example;
Neden gitmedi ki anlamadım. = I did not understand why he did not go.
When we add ki here it sounds like, we wonder a little bit more than average in this sentence.
another example;
O zaten söyledi ki! = He already said! here it just put some more emphasis on the word 'already'.


3. Possesive
Also, we can use it to skip nouns (like mine in English)
benim kitabımı kullan --> use my book
benimkini kullan --> use mine
benimki = mine, seninki = yours, onunki, bizimki, sizinki, onlarınki

and people usually say benimki or seninki referring to their partners in colloquial language.