The "first cousin, once removed" terminology refers to the relationship of individuals who share a common ancestor. Other than your immediate family and direct line ancestors, those related are known as collateral line relatives. While it may sound very confusing, you can typically follow it up to a point -- after that you need the help of a Relationship Chart.
A first cousin, once removed denotes the relationship of an individual to the children of his or her first cousin. That is, the child of your first cousin is your first cousin, once removed. You and your cousin share a set of grandparents. The grandchild of your first cousin, is your first cousin, twice removed. The great-grandchild of your first cousin, if your first cousin, three-times removed.
Second cousins are the children of first cousins. Second cousins share a set of great-grandparents.
Third cousins are the children of second cousins. Third cousins share a set of 2nd great-grandparents.
To determine more complex relationships, you can refer to a Relationship Chart, which allows you to cross-reference relationships stemming from a common ancestor. A relationship chart can help you identify your relationship to others working on your same family line, with whom you have contact.. The Relationship Chart works for any common ancestor, no matter how distant (limited only be the size of sheet on which it is printed). Many relationship charts can be found on the Internet; just enter "genealogy relationship chart" into your search engine, and find the one that works for you.
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