If the grandfathers were 1st cousins, the grandchildren of each grandfather are 3rd cousins to each other.
Third cousins once removed.
Your grandfather's sister's children are your 'first cousins once removed'.
The relationship to the grandfather or the relationship to the grandson of the grandfather? Let's say your name is Ken, your grandfather's name is William, his brother's name is Eli, and Eli's son is named John. If you're asking about William and John's relationship, they are first cousins. If you're asking about Ken and John's relationship, they are first cousins once-removed.
Your grandfather's niece is your parent's first cousin. Her daughter is your second cousins and her granddaughter is your second cousin, once removed.
The children of your great grandfather's brother are your grandparent's first cousins, and your first cousins twice removed.
If two people have the same grandfather, they are siblings or first cousins. If two people have the same great grandfather, they are siblings, first cousins, or second cousins to each other. If two people have the same great great grandfather, they are siblings, first cousins, second cousins, or third cousins.
Two people whose first common ancestor is their great grandfather are second cousins. That describes her and your parent. Since you are one generation further from the common ancestor than she is, the two of you are second cousins, once removed.
Your grandfather's first cousin is your first cousin, twice removed. Your grandfather's second cousin is your second cousin, twice removed. Your grandfather's third cousin is your third cousin, twice removed. The relationship possibilities become more complex if you are talking about your grandfather's cousins who are once (or more) removed.
Since your great grandfather was brother to the other person's great grandfather, the two great grandfathers had at least one parent in common. That parent is your common ancestor, the great great grandparent of both of you. People who share a grandparent are first cousins. People who share a great grandparent are second cousins. People who share a great great grandparent are third cousins. You are third cousins.
If you and someone else have grandmothers who were First Cousins, the parents of those grandmothers were siblings and you and someone else have great-great-grandparents in common. Therefore you and someone else are Third Cousins.
Your great grandfather's nephew was your grandfather's first cousin. You are two generations removed from your grandfather and his cousin. Therefore you and your great grandfather's nephew are first cousins, twice removed.
Second cousins.