The terms 'great cousin' and 'grand cousin' are not ones used in genealogy.
Your family may use the term, but it is not in common usage and has no standard meaning.
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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.