Your first cousin's child is your first cousin's, once removed. Your second cousin's child is your second cousin, once removed. Your third cousin's child is your third cousin, once removed.
Your grandmother's third cousin is your third cousin, twice removed.
Your grandson's third cousin is your first cousin twice removed, and you are hers. You count from the cousin who is closest in generation to the common ancestor in determining what kind of cousin relationship applies.
Third cousin twice removed.
Your cousin's brother is also your cousin. His child is your cousin once removed. For example, if your first cousin has a child, that child is your first cousin, once removed. If your third cousin has a child, the child is your third cousin, once removed.
Your father's first cousin, once removed, is your 2nd cousin. Your father's second cousin, once removed, is your third cousin.
Your first cousin's grandson is your first cousin twice removed. Your second cousin's grandson is your second cousin twice removed. Your third cousin's grandson is your third cousin twice removed. And so forth. If your cousin is once removed, twice removed, etc., the relationship possibilities for you and the grandson become more varied and complex.
Your first cousin's child is your first cousin, once removed. Your second cousin's child is your second cousin, once removed. Your third cousin's child is your third cousin, once removed. Your relationship to the children of cousins who are once, twice, or more removed is more complex and depends on whether you or the removed cousin are in the older generation.
Your first cousin's daughter is your first cousin, once removed. Your second cousin's daughter is your second cousin, once removed. Your third cousin's daughter is your third cousin, once removed. The daughter of your first cousin, once removed, is either you second cousin or your first cousin twice removed. The daughter of your second cousin, once removed, is either your third cousin or your second cousin twice removed. And it gets more complicated from there.
Your grandfather's first cousin is your first cousin, twice removed. Your grandfather's second cousin is your second cousin, twice removed. Your grandfather's third cousin is your third cousin, twice removed. The relationship possibilities become more complex if you are talking about your grandfather's cousins who are once (or more) removed.
Your father's first cousin's grandchild is your second cousin, once removed. Your father's second cousin's grandchild is your third cousin, once removed. Your father's third cousin's grandchild is your fourth cousin, once removed. And so forth.
Your first cousin and your children are first cousin, once removed. Your second cousin and your children are second cousins, once removed. Your third cousins and your children are third cousins, once removed. And so forth.