Two numbers name any blood relationship
English kinship terms look complicated but come from simple counting. Find the nearest ancestor that you and the other person both descend from, then count generations: how many steps up from you to that ancestor, and how many steps up from them. Call those numbers A and B. Every blood relationship in the language falls out of the pair. If A and B are both at least 2, you are cousins: the degree is the smaller number minus one, and the number of times removed is the difference between them.
Worked example: you and another person both descend from the same grandparent. You count two steps up (you to parent to grandparent) and so do they, so A = 2 and B = 2. The smaller number minus one is 1, so you are first cousins, and the difference is zero, so neither of you is removed. Now take that cousin's child. You still count 2 steps to your grandparent, but the child counts 3 to the same person, so A = 2 and B = 3. The degree is still min(2,3) - 1 = 1, and the difference is 1, making them your first cousin once removed.
Removed does not mean distant
The most common misunderstanding is that removed means further away. It only means the two of you sit in different generations. Your first cousin once removed is either your first cousin's child or your parent's first cousin, and both share about 6.25 percent of your DNA, while a second cousin in your own generation shares only about 3.125 percent. Second cousins are the grandchildren of siblings; third cousins are the great-grandchildren of siblings.
When one of the numbers is 0 or 1, the language switches from cousins to direct terms. A = 0 means the person descends from you: child, grandchild, great-grandchild. B = 0 means they are your ancestor: parent, grandparent, great-grandparent. A = 1 and B = 1 makes you siblings. A = 1 with B larger gives nephew or niece, then grand-nephew, then great-grand-nephew. B = 1 with A larger gives aunt or uncle, then great-aunt, then great-great-aunt. Genealogists often write grandaunt where everyday speech says great-aunt; the two mean exactly the same person.
Shared DNA and the fine print
Expected shared DNA halves with each generation of separation. Full siblings average 50 percent, aunt and nephew 25 percent, first cousins 12.5 percent, first cousins once removed 6.25 percent, second cousins 3.125 percent, and third cousins about 0.78 percent. These are averages of a random process, so real testing results vary widely, and beyond third cousins two people may share no detectable DNA at all while still being related on paper.
The calculator assumes full-blood relationships through both members of an ancestral couple. Half siblings share one parent rather than two, so every figure halves: half first cousins average about 6.25 percent. Step and adoptive relationships carry the same social names but no shared ancestry. English also flattens distinctions that other languages keep: one word, aunt, covers a father's sister, a mother's sister and an uncle's wife, and cousin covers both sides and both sexes.