Yes, your niece's daughter is your grand niece. You are her great aunt or great uncle.
Your niece's daughter may be called either your grand niece or your great niece. You would be her great aunt or great uncle.
The daughter of your grand niece (some would call her your great niece) is your great grand niece (some would call her your great great niece).
Yes, your sister's daughter is your niece and you are her aunt (if you are female) or her uncle( if you are male).
Yes, your great niece's daughter is your great-great niece.
She is your great-grand niece.
It is a great grand neice.
Grand
Your niece's daughter is your grand niece, or, as some would say, your great niece.
Grand niece.
You are her grand uncle (or great aunt) and she is your grand niece.
Your niece's daughter is your great niece. You are her great aunt or great uncle, depending on your sex.
My niece's daughter is my great niece.
Great-niece.
You are her great-great Aunt (or great grand Aunt, depending upon your preferred nomenclature). She is your great great niece.
Your nephew's child is your great niece (if a girl) or your great nephew (if a boy).
your 2nd niece...!
Your daughter and your niece are first cousins Your daughter and your great grand niece are first cousins, twice removed.
Your niece's son is your grand-nephew. You are the child's grand aunt, or grand uncle (whichever applies).
Your niece's son's daughter is your great grand niece. Your niece's son is your grand nephew or your great nephew, whichever term you prefer.