I know that my parents' siblings' child is my cousin.
But what about the grandchild of my parents' siblings?
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The naming of different types of cousins is referred to as “ordinals and removals”.
If you share the same grandparents, you are “first cousins”. If you share the same great grandparents, you are “second cousins”. And so on.
The children of your first cousin are your “first cousins once removed”. The grandchildren of your first cousin are “first cousins twice removed”. The children of your second cousin are your “second cousins once removed”. And so on.
Note that most people have no idea how this system works and just use the generic “cousin” for all distant relatives.
The grandchild of your parent's sibling is your first cousin once removed. (You are also their first cousin once removed.)
Here's another attempt at a diagram, simply because this form works better for me, maybe it will for you too. The number of "removals" are the extra generations on one side only (shown in orange below), and once those extra generations are removed, the degree indicates how many more generations to get back to siblings (shown in blue below).
The standard family tree chart never made a lot of sense to me, so I drew the following "ladder picture", where first cousins, seconds cousins, third cousins, and so forth, are rungs on a ladder:
This visual might be easier for some folks to understand. To answer your specific question:
what about the grandchild of my parents' siblings
Person 3 and Person 6 are first cousins once removed.
The child of an aunt or uncle is your
The grandchild of an aunt or uncle is your
Please see the chart at What Is a Second Cousin?
Wikipedia has some good stuff on this, e.g.
My German wife had an awful lot of aunts (Tanten) and uncles (Onkel), whereas I have a lot of cousins. It wasn't until I saw these diagrams that I understood why.