Your first cousin's child is your first cousin once removed.
Your children and that child will be second cousins to each other.
This poster above is incorrect. If you have the same grandparents as someone on either side, they are your first cousins. Your first cousins once removed would be if your great-grandparents were someone else's grandparents. Your second cousins are if you have the same great grandparents as someone. etc.
You are first cousins once removed to each other.
You and the child of your first cousin are first cousins once removed.You and your first cousin's child are first cousins, once removed, to each other.
Your cousin (if a first cousin) and your child are first cousins once removed.
Your first cousin's child would be your second cousin
Your child and your first cousin are first cousins, once removed to each other.
Your child and your first cousin's child are second cousins. A full generation is another set of cousins. For example, the second cousins' children will be third cousins, etc. Now here is the tricky part. You are a first cousin once removed to your cousin's child and your cousin is a first cousin once removed from your child.
Your first cousin and your grandchildren (your child's children) are first cousins once removed.
Your child and your sister's children are first cousins. Your grandchild and your sister's children are first cousins once removed.
You and your uncle's child are first cousins to each other. You and your uncle's grandchild are first cousins, once removed, to each other.
Assuming both children are on the same side of the family, your child's first cousin is a second cousin to your first cousin's child. Your child is also a second cousin to your first cousin's child. Your child can have first cousins who are not related to your first cousin and thus not related to your first cousin's child.
Your first cousin's child is your child's second cousin. Your second cousin's child is your child's third cousin. Your third cousin's child is your child's fourth cousin. and so forth
Your child and your first cousin's child are second cousins. Your child and your second cousin's child are third cousins to each other.