The answers lie in the generations. It begins with children. The children of siblings are first cousins because they are the first generation of cousins. The children of first cousins are second cousins to one another. And so on through the generations.
"Once removed" doesn't indicate that a cousin was ever banished from the family—just that the cousins are separated by a generation. So when your first cousin has a child, that child isn't your niece or nephew, so what are they to you? You and that child are first cousins once removed. Your children and your cousin's children are second cousins. When your first cousin has a grandchild, that grandchild is your first cousin twice removed, because they are two generations from you. But by the time people reach grandparenthood, almost everyone will probably be introducing one another as "cousin," skipping the twices and removes, which is both polite and correct—and easier.