Your grandson and your sister's grandson are second cousins to each other.
They would be the fourth cousins. They share a common 5 Great Grandparent.
The grandson of your first cousin and your grandson are third cousins to eachother.
Your grandson and great grand son could be cousins,(first cousins once removed), they could be uncle and nephew, or they could be father and son. It depends on how they are related to your children. If your great grandson is the grandchild on one of your children and your grandson is the child of a different one of your children, they are first cousins, once removed. If your great grandson is the child of a sibling of your grandson, then they are uncle and nephew. But your great grandson could also be the son of your grandson. Then they are father and son
Your grandson and your brother's grandson are second cousins to each other.
Your granddaughter and your first cousin's grandson are third cousins to each other. You find their common ancestor that is, your grand parents, and count the number of generations separating them, there are Brother and Sister then cousins, you and your cousin would be one (first cousins), your children and your cousins children are second cousins and their children are third cousins and so on.
Your niece and one of your grandson's parents are first cousins. That makes your niece a first cousin once removed of your grandson.
Your niece's child and your grandson's child (your great grandchild) are second cousins, once removed, to each other.
Your grandson and your first cousin are first cousins, twice removed. Your grandson and your second cousin are second cousins, twice removed. and so forth.
Your first cousin's daughter is first cousin once removed to your grandson, and he to her. Your second cousin's daughter is second cousin once removed to your grandson, and he to her.
Second cousins.
Great nephew. Your children are the first cousins of your niece and your children would be first cousins once removed. Your grandchildren and your niece's child would be second cousins.