Wednesday, Unmitigated (AKA one of my sisters) asked me to comment on what makes cousins beyond first and to explain the "removed" part. I will try my best.
John & Mary Percival have sons, we will call them John and Frank, get married and have children. John has 3 children and Frank has six. The children are first cousins or "cousins."
Frank's oldest daughter has a daughter; John's oldest daughter does too. These new additions to the family are second cousins to each other, but John's granddaughter is a first cousin once removed (a generation removed) to Frank's children. The same for Frank's new granddaughter, she is a first cousin once removed to John's children.
Skip ahead 20+ years and those grandchildren of John and Frank have children. They are third cousins to each other, but John's great grandchildren are 1st cousins twice removed from Frank's children and 1st cousins once removed from Frank's grandchildren.
To put it more succinctly, the original set of first cousins are 1st generation, their children are second cousins, their grandchildren are third cousins, etc. The removed part is the generation "removed" from the original generation of cousins. Click on the image to see a larger view.
Copyright 2010, ACK for Gene Notes