'Third cousin' refers to how many generations you are from your common ancestor. Third cousin means that the two people share the same great-great-grandparents. Another way of looking at third cousin is to know that two people who are third cousins are the great grandchildren of siblings.
The 'once removed' part refers to how many generations different is your distance than your cousin's from the common ancestor. If two people are third cousins, then one of those two and the child of the other are third cousins, once removed, because the child is one generation further removed from the common ancestor.
Either:
Your third cousin twice removed is the grandchild of your third cousin. That person could also be described as the great great great grandchild of the brother or sister of your great grandparent.
In addition, your third cousin twice removed can be the third cousin of one of your grandparents. That person could also be described as the great grandchild of the brother or sister of one of your great great great grandparents
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 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 grandmother's third cousin is your third cousin, twice removed.
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 first cousin's granddaughter is your first cousin, twice removed. Your second cousin's granddaughter is your second cousin, twice removed. Your third cousin's granddaughter is your third cousin, twice removed. And so forth...
Your grandmother's second cousin is also your second cousin, but twice removed. The twice removed means the number of generations between you. Your mother's first cousin is your first cousin once removed. If your mother's first cousin has children, they are your second cousins.
Your first cousin's grandsons are your first cousins, twice removed.
Your mother's second cousin twice removed is either your third cousin once removed or your second cousin three times removed. Which it is depends on whether the second cousin twice removed is the grandchild of your mother's second cousin or the grandparent of her fourth cousin.
The children of your first cousin once removed are your first cousins, twice removed, or they are your second cousins, depending. The children of your second cousin once removed are your second cousins, twice removed, or they are your third cousins, depending.
The child of your 2nd cousin once removed is your 2nd cousin twice removed if you are in the older generation; or your third cousin if you are in the younger 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.