The relationship of a "third cousin twice removed" is that of third cousins who are two generations apart. A third cousin twice removed can be the grandchild of a third cousin or the third cousin of a grandparent. Third cousins can also be the grandchildren of two first cousins. Thus, if John's grandfather and Joe’s grandfather are first cousins, John and Joe are third cousins.
Third cousins can also be the children of two second cousins. Second cousins are the children of first cousins. Thus, if John's mother and Joe’s mother are second cousins, John and Joe are third cousins.
Third cousins share great-great-grandparents but not great-grandparents, grandparents or parents.
The term "removed" indicates that the two people are of different generations. "Once removed" indicates a difference of one generation, and "twice removed" indicates a difference of two generations.
Put another way, if Jack’s great-great-grandparent is Jill’s great-great-great-great-grandparent, Jack and Jill are third cousins twice removed. This is true only if Jack and Jill are not direct descendants, which means that Jack is not Jill’s great-grandfather.
The terms "first cousin," "second cousin," "third cousin" and so on, refer to the number of generations between two people’s closest common ancestor. The terms "once removed," "twice removed" and so on, refer to the number of generations between the two people themselves.