Cecil was correct in his assessment of what to call your cousin’s children (they are your 1st cousins, once removed). Your cousin’s children’s children are you 1st cousins, twice removed, and so on.
However, there’s another interesting way to think about it. The definition of what makes someone your cousin is that the two of you share grandparents. Similarly, you share great-grandparents with your 2nd cousins, great-great-grandparents with your 3rd cousins, and so on. When you throw the removal into the mix, we find that with your 1st cousins, once removed, your grandparents are their great-grandparents (as they are also your cousin’s grandparents). So the “degree” of cousin is a measure of what set of gradparents you share, and the degree of removal is the shift in generations.
From this we can extrapolate in the other direction as well. If you share great-parents with your 2nd cousins and grandparents with your 1st cousins, then it follows that you share parents with your 0th cousins. Thus, your brothers and sisters are also your 0th cousins, and your nieces and nephews are your 0th cousins, once removed. Notice also that your children and your brother’s or sister’s children, being of the same generation, would gain a degree of cousin and lose a degree of removal, thus becomming 1st cousins, exactly as it should be.
It can further be deduced from this that you are your own -1st cousin.
In Norwegian your cousins are either ‘søskenbarn’ (siblings-children - with the sibling ‘obviously’ refering to your parents’ siblings) or ‘fetter’ (male) and kusine (female). Numbering comes into play with your 2nd cousins, tremenning (tre = three), 3rd cousins, firmenning (fire = four) and so on. There’s no ‘removed’, you’re forced to use more lengthy and IMO more comprehendible counstructions like ‘my cousin’s children’, ‘my third cousins’ parents’.
The astute reader will have noticed that the number part doesn’t match, which means you have to pay attention when you do genealogy and translate between Xth cousin and Z-menning. (X = Z-1)
To make things worse for us genealogists there’s a number mismatch in great-great’s as well.
Grandparents = besteforeldre
Great-grandparents = oldeforeldre
Great-great-grandparents = tipp-oldeforeldre
Great3-grandparents = Tipp2-oldeforeldre
GreatX-grandparents = TippZ-oldeforeldre (with X=Z+1)
Mix up cousins and grandparents and all of a sudden you’re short a couple of generations.
With my Chinese family, what matters is the generational relationship.
So my mom’s male cousin is my “uncle-cousin”, usually just contracted to “uncle”, because he’s considered equivalent to my mom’s brother.
Due to the largeness of my family and the various ages at which they’ve gotten married and have kids, this has resulted in me having “aunts” and “uncles” that are occasionally younger than I am!
With my Korean family, they basically count the number of people involved. So between my cousin (sa-chon) and myself, there is me (1), my father/mother (2), their brother/sister (3), and their kid (4). Sa is 4 in Korean. So for my cousin’s kid, it’d be o-chon, or five-chon, since you’ve added another to the mix. Uncle is sam-chon, or 3-chon. Then they had to confused things by adding in titles based on what side of the family, and gender into the mix. But it’s really quite simple at the heart of it.
As to the modes of address, Cecil says that just “cousin” for a first cousin once removed would be awkward because of the difference in ages, but I’m not sure that’s necessarily so. My oldest first-cousin-once-removed-down is only three or four years younger than me, and I have a dozen full cousins who are younger than her. Meanwhile, my oldest full cousin (father, incidentally, of my oldest 1st once removed) is about twenty years older than me. If age were an obstacle to calling someone “cousin”, then I shouldn’t be calling my eldest cousin “cousin”, but I should call his daughter that.
In my family, therefore, we avoid the problem by just referring to everyone (aside from ancestors, decendants, siblings of ancestors, and decendants of siblings) as “cousin”, in casual conversation, and as “nth cousin m times removed” when greater precision is called for (and yes, we do all understand the cousin terminology system: In a family the size of ours, it’s a necessary survival trait).
My family’s not all that big (and it’s positively tiny for a Catholic family). There is a certain understood formalism for references, though. (slight hijack here).
In the main line beyond the parents, referring up it’s the relationship followed by the last name. Thus: “Grandma and Grandpa Armstrong” and “Grandma and Grandpa Nichols”. With parents and in conversation it becomes simply the relationship. “Mom”, "Dad, “Grandma”, “Grandpa”, and so on. Of course, for me this is moot since the main line now ends at my parents.
In the main line referring downwards it’s the name. I’m always John to them, possibly “my <relationship> John” if referring to me in conversation with someone else.
Off the main line as a descriptor is the full formal relationship followed by the given name. I have “my uncle Jim”, “my aunt Mary”, “my cousins Patty and Paul”, “my first-cousin once-removed Danille”, “my second cousins Max and Jack” “my great-uncle Lad”, and so on.
In conversation the references are usually flattened if the person I’m talking to knows the other person. In conversation with my uncle Jim I’ll refer to “Cousin Danille” or “Uncle Lad”, since he knows to expand it. In conversation with the person himself, the first exchange is semi-formal (“Hello, Uncle Jim”, “Hi Cousin Danille”), but after that it’s just the first name. With uncles and aunts (and great-uncles and so on) there’s a slight variation: they always refer to me as “John”, partly out of their superior position and partly because “Nephew John” is a little awkward. With extended cousins it’s still “Cousin John” at the opening and “John” thereafter.
I’m sure there’s a sociology paper in here somewhere.
I refer to my cousins exclusively by their names or, very rarely, an ironically intended “cuz.” There’s typically no distinction made between cousins of different orders when talking about them with third persons – they’re all cousins unless greater specificity is called for, and why would it be? I only really knew my paternal grandparents, so they were always Granma and Grampa. Uncles and aunts (whether actual or great) are typically “Uncle Steve,” etc., on first reference, then just Steve (or whatever) thereafter. The exception is my Aunt Muriel and Uncle Marvin (before his passing last year). They live/d nearby and I see them more often than anyone else in my family (no one else lives closer than six hours away), so we’ve gotten in the habit of merely using their names.
Huh. I had once, long ago, come to the conclusion that I’m my own -1st cousin, but -2nd cousins is nerdier than even I went. There’s a part of me that wants to say that’s wrong, but I can’t say why.