Cousin names look like a puzzle until you see that they come from two counts. Find the nearest ancestor you share with the other person. Count how many generations that ancestor is above you, and how many above them. Call these a (yours) and b (theirs).
The degree
Take the smaller of a and b and subtract one. If you both sit two generations below your shared grandparents, a and b are 2 and 2, and the degree is 1: you are first cousins. If you are three below your shared great-grandparents, the degree is 2: second cousins. Siblings are the case where a and b are both 1, so there is no cousin degree at all.
The removal
If a and b are equal, the cousins are of the same generation. If they differ, the difference is the number of times removed. Your first cousin's child sits one generation lower: a is 2, b is 3, so they are your first cousin once removed. Their own child is twice removed, and so on.
Which way round?
The word "removed" does not say whether the other person is older or younger. Your first cousin's child is your first cousin once removed, and you are that child's first cousin once removed too. The names are mirror images, which is why most cousin terms are their own reciprocal.
A common slip
Your parent's first cousin is not your second cousin. The shared ancestors are your great-grandparents, but you are three steps away and your parent's cousin only two, so the pair is a=3, b=2: first cousin once removed. Second cousins share great-grandparents and sit at the same level.
Try it with the calculator, look up a result on the chart, or read the page for first cousin once removed.