It's quite different Susan. The removed defines your relationship to cousins of your parents/grandparents etc.
In the previous example - If your new relative was the 4th cousin of your mother/father, you would be the 4th cousin once removed.
I know quite a number of my mother's 1st cousins. I am the 2nd cousin to their children, but the 1st cousin once removed to them.
My daughter is the 1st cousin twice removed to my mother's cousins (her grandmother's cousins) and the 2nd cousin once removed to their children (her mother's 2nd cousins).
Trish
I should add, given that our extended family is quite large - we just call everybody cousin
