I'm confused about the "country divisions" in the following. Does "country" there refer to "a state" or "a rural area"? What kind of division is it? And what's the difference between a division and a department in a company or an organization?
Big organizations especially have complex matrix arrangements, for example, country divisions, product divisions, and company-wide corporate departments like human resources, all at the same time.
They mean a nation. One way of dividing up a large, multi-national company is to do it by country.
Other ways are by function or product line. A "matrix" organization recognizes that people do have different locations in different interlocking webs. They are not just strictly in one hierarchy of corporate organization.
You might be an accountant for X product line in Y country.
"Departments" are usually smaller than "divisions," but the names can be interchangeable as well.