I've never been very good at the whole degrees-of-family thing. I've never really bothered to figure it out, and whenever someone would say 'so-and-so is a third cousin thrice removed' I'd just nod. I'm lazy that way.
This week, however, I got a friend request on Facebook from a distant family member, whose branch of the family moved to Canada many years ago. I was chatting to my aunt about it and she said 'she'll be your second cousin'. This didn't feel right, because on my father's side of the family, most of my cousins were my parents' age, and their children were our age, and we always referred to them as our second cousins.
Feeling rather ashamed at my lack of mastery of this whole degrees-of-cousins thing, I decided to educate myself (thanks Wikipedia) and put an end to my ignorance, once and for all.
As it turns out, my aunt was correct, but calling your cousin's children your second cousins is a common (but incorrect) usage of the term, and they're more strictly first cousins once removed. Overall, it works like this:
A system of degrees and removes is used to describe the relationship between the two cousins and the ancestor they have in common. The degree (first, second, third cousin, etc.) indicates the minimum number of generations between either cousin and the nearest common ancestor; the remove (once removed, twice removed, etc.) indicates the number of generations, if any, separating the two cousins from each other.
Wikipedia has a pretty graph which sort-of explains it better. The trick of it is that because the degree refers to the minimum generations between either you or your cousin, and the common ancestor, you can have a first cousin once removed no matter far out the common ancestor goes. Between your nth cousin and the common ancestor, moving up the generations reduces the degree and increases the removed.
Did that make sense? I'll probably have forgotten it by tomorrow as well.