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.
There is no difference between "grandniece" and great niece." Some people prefer to use "grand" but others prefer to use "great" when talking about the granddaughter of one of their siblings.
The son of your great grandfather's brother was your grandfather's first cousin. That makes him your first cousin, twice removed.
Your ancestors are the people whose children lead eventually to you. They are your mother and father, grandmothers and grandfathers, great grandparents, great great grandparents, etc. Your cousins are the children, grandchildren, great grandchildren, etc, of the brothers and sisters of your ancestors.
== == When people talk about their cousins, 'removed' is a reference to the difference in generations. Your mother's cousin would be your cousin once-removed. The children of your mother's cousin would be your second cousins, the same generation of descendants from the common ancestors. Most people just refer to them as cousins and don't worry about the specific generation. The genealogists in the family like to be more specific.
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 grand-niece.
Your first cousin's great grand uncle (also called great great uncle) could be your great great uncle, or could be unrelated to you. The same applies to the great great uncle of your second cousin. For other cousin relationships the possibilities become more varied, but always include not being related to you at all.
Your great aunt's son is your parent's first cousin and your first cousin, once removed.
second cousin once removed
Your nephew is the son of your sibling and a first cousin of your childrenYour grand nephew (or great nephew) is the grandson of your sibling and a second cousin of your grandsonYour great great nephew (or great grand nephew) is the great grandson of your sibling and is a 2nd cousin once removed of your grandson
Your mother and her second cousin share a common great grandparent. There is a difference on one generation between the two of you. That would make them a second cousin once removed.
There is no difference between "grandniece" and great niece." Some people prefer to use "grand" but others prefer to use "great" when talking about the granddaughter of one of their siblings.
Since the ancestors of your great uncle's wife are not related to you, the first cousin 14 times removed of the wife of her 15th great grand uncle is not related to you, nor you to him.
Second cousin, twice removed
You and your second cousin share a common great grandparent. There is a difference of two generation between the grandchild and you. That would make you second cousins twice removed.
While some people prefer the terms grand aunt or grand uncle, parallel to grandmother or grandfather, the terms great aunt and great uncle are widely used and accepted, and are functionally the same.Your great aunt is the sister of one of your grandparents, which makes you a grand niece or grand nephew (great niece or great nephew). The son of said grand aunt is a first cousin of one of your parents. The son's daughter is first cousin once removed to your parent, and second cousin to you.