Second cousins would be your cousin's cousins. It basically comes down to which level of grandparent you have in common. Grandparents = cousins, great-grandparents = second cousins, great-great-grandparents = third cousins.
The "removed" refers to differences in generational cohort. So if you are connected to the person by your grandparent but that same person is a great-grandparent to the relative, that relative is once removed from you. So your first cousins' children are first cousins once removed. Their grandchildren would be your first cousins twice-removed.
The "removed" refers to differences in generational cohort. So if you are connected to the person by your grandparent but that same person is a great-grandparent to the relative, that relative is once removed from you. So your first cousins' children are first cousins once removed. Their grandchildren would be your first cousins twice-removed.