There are several possibilities, including "no relation" (or at least none that can be described more concisely than "great uncle's niece").
Assuming the maximum possible degree of consanguinity, your great uncle's niece could be your mother. She could also be an aunt (your mother's or father's sister, depending on which side the great uncle is on), or as mentioned she could be "no relation" (if the great uncle is an uncle by marriage and his niece is the daughter of one of his brothers or sisters).
Your great-grandparents' nephew would be your 1st cousin twice removed. See discussion for explanation.
If your Uncle is one of your parents siblings then his Great Grandparents are your great-great grandparents.
Your first cousin once removed.
A niece's son calls his uncle his uncle, just like everyone else.The niece's son calls her uncle his great uncle. He is the great uncle's great nephew.
Your niece's son is your great nephew. You are his great aunt or great uncle.
You are your nephew's uncle, or aunt, and his son's great uncle, or great aunt. Your nephew's son is your great nephew.
Your sister's son is your nephew. Your uncle is your nephew's grand uncle or great uncle.
Your mother's brother is your uncle. Your uncle is your son's great uncle. You are the nephew of your mother's brother. Your son is his great nephew.
His great uncle
You are the grand uncle to any children your nephew or niece have.They would be your grand nephew or grand niece.If the children of your nephew/niece have children of their own, you'd be a great uncle too.
Great uncle / Great nephew
Your Great Uncle's son would be your 2nd cousin.
The son of your great niece is your great great nephew. You are his great great uncle.
your uncle is your sons great uncle
Your nephew's grand son is your great great nephew and you are his great great uncle (or aunt)