Herkese Merhaba,

Annem içeri giriyor
My mother is coming in.

Is it correct to say?
Annem içeriye girior.

(and If so, then what is the difference between içeri and içeriye?)

içeri (an adverb) means inwards or into

-eri is a suffix for direction and can be found in some words:

içeri, dışarı, yukarı, ileri, beri, geri, aşağı (from altarı)

so içeri giriyor = going in (or going in the direction towards the inside of something)

içeriye giriyor is ok. In that içeri (as a noun) means inside so it means

going to the inside.

Thank you very much for the detailed clarification!