Explaining cousins

Cousins have common grandparents. Second cousins have common great-grandparents. Third cousins have common great-great-grandparents, and so on.

Now, if cousins have common ancestors, but those ancestors do not sustain the same relationship to both cousins, then the "removed" label comes in. If you and I are cousins, but our common ancestors are my grandparents and your great-grandparents, then we are one generation removed from being first cousins. We cannot be second cousins because our ancestors are not my great-grandparents, and we cannot be first cousins because our ancestors are not your grandparents. So, we use the lowest-common denominator (in this case first cousins once removed). If we share only one common ancestor (we have common grandfathers, but not common grandmothers, for instance), then we are half-first cousins.