"Мне" is a dative case of "Я" ("I" or "Me") and means "To me". It's used independently from the gender of speaker. For example: Он дал это МНЕ - He gave it TO ME.
You can say "This is MY house" as "Это МОЙ дом".
"Мой" is a possesive adjective and stands for "My", which is changed due to the gender of the object, it is applied to. For example: МОЙ дом (MY house - masculine), МОЯ жизнь (MY life - feminine), МОЁ мнение (MY opinion - neutral), etc.