"Neha did not come to my home."
Translation:नेहा मेरे घर नहीं आयी।
The form मेरे is in the oblique case possessive form and मेरा is the direct case possessive form. Learning the oblique case is essential part of learning Hindi. Oblique Case has many uses. You just saw one of them.
For one, the oblique case is used when a word is the object of a postposition. E.g. लड़का (boy) → लड़के के साथ (with the boy). Here लड़के (oblique case of लड़का) is the object of the postposition के
why is आयी? The Past Feminine Singular form of the verb आना is आई.
For verb roots that end with a vowel we add या yaa for the Masculine Singular case but all other cases are the same.