I think the charts provided by AncestryDNA when you have a match with someone who also has a common ancestor listed in their tree are a great way to see this "removed" business at work.
If you look at your DNA matches and you see a little leaf it means you and that match have someone in common listed in your tree resulting in a "shared ancestor hint."
Here is an example of someone and I having my forth great grandparents in common; Leonard L. Losee and Lydia Ann Smith-Losee.
Moving down the tree:
- John M. Losee Sr. and Sarah Elizabeth Losee-Combs were siblings.
- John M. Losee Jr. and Stella Combs were first cousins.
- Ethel Mae Losee-Earle and the first male listed as Private on the right would have been second cousins.
- My Grandpa Earle (the first male listed as Private on the left) would have been third cousins with the male directly across from him on the right.
- My father and the male directly across from him would be 4th cousins.
- If there were anyone across from my that person would be my 5th cousin.
This chart indicates my relationship to each person presented. See that (1x removed). That man is my father's 4th cousin; he is my 4th cousin once removed because I am one generation away from him. The man who was my grandfather's third cousin would be my third cousin too just twice removed or the two generations i am distant from him; my father being one generation away, my grandpa two generations away.
So NO my father's cousin is not my second cousin. My father's first cousin is my first cousin...once removed.
My father's first cousins' children are my second cousins. Their kids are my second cousins once removed.