An example..
Egypt was "under the British" in the later part of the 1800s until the mid 1900s.
What does that mean or imply? Does it affect culture? politics? currency?
I don't get the concept itself or how it works. For instance, what changed in Egypt when it was "under" the British? What did that mean for Egypt?
What are the implications of being "under" an Empire or country?
Well, specifically in the case of Egypt it was "under" a British civil and military occupation from 1882 until Egypt's nominal independence in 1922 and that arguable continued (albeit to a much lesser degree) until Nasser threw out the last vestiges of British imperial rule when he nationalized the Suez Canal in 1956.
In practical terms this meant that during that period from 1882 to 1922 Egypt ceased to be an independent country. Its foreign and domestic politics were determined by British and not Egyptian administrators. This had wide reaching effects on virtually all aspects of Egyptian life including, as you mention, culture, politics and economics.
edit: word