Which part of speech and which kind is delhi?

The Word "Delhi" (capital D) is a proper noun as the name of a specific territory in India, the capital of that territory, New Delhi; a town in New York State and a town in Ontario Canada; the name of a place.

A proper noun is the name of a specific person, place, or thing.

A proper noun is always capitalized.

A homophone (a Word pronounced the same) for the proper noun "Delhi" is the common noun "deli" (lower case d), a shortened form of the noun "delicatessen", a Word for a type of store that sells ready-to-eat foods; a Word for a thing.

A common noun is capitalized only when it is the first Word in a sentence.