"غيري " means "other than me".
For example: lan tajida a7adan ghayrii ya9balu bik = You won't find someone other than me who accepts you.
Edit: Also, you should say "wa kayfa tusta5dam" not "wa kayfa yasta5dimha".
1) As an additional bit of information appertaining to HBZ55's correction "Also, you should say 'wa kayfa tusta5dam' not 'wa kayfa yasta5dimha'", allow me to mention that "tusta5dam" is the singular feminine conjugation of a closely related verb which is constructed in the passive. The literal meaning is "she is used", but we would translate it into "it is used". It's feminine because it refers back to the word "kalima", which is grammatically feminine. Your use of "yasta5dimha" means "he use her/it".
Your sentence "ma ma3na alkaalima "غيري " wa kayfa yastakhdimha?" means "What is the meaning of the word "ghairy" and how does he/it use her/it?" I am sure you meant what HBZ55 inferred when he made his suggestion "wa kayfa tusta5dam". His substitution changes the meaning of your sentence into "What is the meaning of the word "ghairy" and how is she/it used?"
2) "غيري" (ghairy) is a combination of two elements. "غير" (ghair) means "other". "ي" (pronounced as a long e, as used at the end of "غيري") means "me", and is the attached form of the independent pronoun "أنا" "ana", which corresponds to English "I".
"غير" is a common noun in both MSA and colloquial Arabic. You can say "غير ذلك" (ghair dhalika) to mean "other than that".
If you double the middle consonant of "غير" (ghair) (the middle consonant is "ي", the letter "yaa") you get the verb "غيَر "
(ghaiyyara), which is conjugated in the past tense for exactly one male individual. It means "He changed", and it is necessary to assign a direct object to this verb. For instance, you can say "غيَر اسمه" (ghaiyyara ismahu) to mean "he changed his name". The "he" in "he changed" is built into the verb. "اسم" (ism) means "name" and "ه" (hu) means "his". "ه" (hu) is the attached pronoun corresponding to the independent pronoun "هو" (huwa). The difference between the two forms is that "ه" (hu) means "his" while "هو" (huwa) means "he".
The use of غير in colloquial forms differs from place to place, and is a bit complicated to explain.
It's unfortunate that the OP didn't provide a context for this term. It's also worth noting that (غيريّ), or 'ghayriyy', can function as an adjective meaning 'altruistic/selfless'. AIUI, it comes from the root noun 'ghayr', or 'other (person/people)', plus the suffix '-iyy', which forms relational adjectives.