A nephew is a cousin to his aunt's son.
Your nephew's son is your great nephew.
great nephew
If your half-sister had a son that would be a half-nephew to you
You would be a great uncle, so they would be your great nephew
If your niece's is by your sister but the brother is by your sisters new husband & another woman there would be no blood relation. If not then he's the nephew
You would be his Great Uncle and he would be your Great Nephew.
The sister's son would be the nephew of her brother.
Your nephew's daughter's son is your great-great nephew. Your nephew's daughter is your great niece.
He would be a nephew, unless of course he is a son from a previous marriage, then he would be no relation.
Your nephew.
your second nephew
Your nephew and your daughter are first cousins to each other. Your nephew's son and your daughter are first cousins, once removed. Your nephew's son and your daughter's child would be second cousins to each other.