In some parts of the English-speaking world the term for the children of your niece is great nephew and great niece. In other parts of the English-speaking world they say grand nephew and grand niece. Either form is "acceptable" somewhere. Use the form you feel comfortable with. If you find many people around you are using the other form, whichever that may be, and you want to "fit in" feel free to use the other form, or not, as you prefer.
Your nephew's child is your great niece (if a girl) or your great nephew (if a boy).
Your nephew's children are your great nephews and great nieces (or grand nephews and grand nieces).
The son of a great grandson is a great greatgrandson.
The children of your grand nephew (or great nephew) are your great great nephews and great great nieces (depending on whether they are girls or boys).
There is no special term in the English language to describe a relationship between you and your nephew's wife. To her, you are just "my husband's uncle." If she has children with your nephew, they will be your great nephews and great nieces and you will be their great uncle or great aunt.
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.
Some would call your nephew's children your grand nephews and grand nieces, others would call then your great nephews and great nieces.
There is no difference between a great nephew and a grand nephew. In some places "great nephew" is the preferred term for the grandson of one of your siblings. In other palces, grand nephew" is the preferred term for that relationship.
Your nephew's son is your grand nephew or great nephew.
Your nephew's grand son is your great great nephew and you are his great great uncle (or aunt)
Your nephew's baby would be your grand-nephew or grand-niece. You would be their grand-uncle or grand-aunt.
Your great great nephew's child is your great great great niece or great great great nephew, depending on the sex of the child.