From a common ancestor, these are common kinship terms for the first four generations.
First cousins share a set of grandparents. They’re the children of siblings.
Second cousins share the same great-grandparents. They’re the children of first cousins.
Third cousins share the same great-great-grandparents. They’re the children of second cousins. First, second, and third cousins (and so on) are the same number of generations removed from the common ancestor (the grandparent) as one another (for example, first cousins are both two generations removed from the grandparents they share, and second cousins are both three generations removed from the grandparents they share).
In cousin relationships, the term removed indicates the separation of a generation. Your first cousin is the same generation as you are, so your first cousin once removed would be either your parent’s first cousin or your first cousin’s child. Since twice removed indicates a difference of two generations, your first cousin twice removed would be either your grandparent’s first cousin or the grandchild of your first cousin (as you are two generations younger than your grandparent’s first cousin and two generations older than your first cousin’s grandchild).
Great and grand
Like removed, the terms great and grand indicate the separation of generations. While “removed” is used for cousins, “grand” and “great” are used both for grandparent/grandchild relationships and for aunt and uncle/niece and nephew relationships.
With grandparents, grandchildren, aunts and uncles, and nieces and nephews, the generation two generations away gets the designation grand, while further generations have great tacked on. For example, your grandfather is two generations away from you, and your great-grandfather is three generations away. Your grand-niece (the child of your niece or nephew) is two generations away from you, and your great-grand-niece (the grandchild of your niece or nephew) is three generations away.
People are half siblings when they share one biological parent and not the other. All other half relationships stem from an original half sibling relationship. Your half nephew is the son of your half sibling; your half great-aunt is the grandparent of your parent’s half sibling; the children of half siblings are half cousins. Since only people related to you through your half sibling can be “half” relationships to you, your direct-line ancestors and descendants can only be “whole” relationships to you.