This is the 17363rd most frequent Russian word.
твоего
"Твоего" means "your" (possessive, singular, and genitive case).
The word 'твоего' is the masculine singular genitive form of 'твой,' used as a possessive adjective referring to the second person singular; here it modifies 'друга' (friend), showing possession.
Я видел твоего друга вчера.
I saw your friend yesterday.
'Твоего' is used as the masculine singular accusative form to agree with 'кота' (cat), indicating that the cat belongs to the person being addressed.
Я нашёл твоего пропавшего кота.
I found your missing cat.
In this sentence, 'твоего' serves its genitive masculine singular form to align with 'отца' (father), indicating a relationship of possession or belonging to the person addressed.