"imperial core" isn't a phrase we made up. It refers to World Systems Theory, a theory of international relations invented by a guy named Immanuel Wallerstein which argues that imperial "Core" countries (think the traditional "developed" or "first world" countries. Mainly the US and Europe) have a particular extractive, colonial relationship with "Periphery" countries (think poor, raw material exporting, rentier states like Kyrgyzstan or Nigeria).

Then there are semi-periphery countries which are still tied into the imperial core in some way, but have enough sway economically and geopolitically to kind of stand on their own. They have a different kind of relationship to the imperial core, compared to the periphery (these would be the BRICS countries, largely).

That's a gross over simplification, but hopefully that answers your question.

Edit: Here's a really good explanation of World Systems Theory that goes into more depth

It's another term for 'the West' effectively, so the US and Western Europe.

The imperial core is the countries that have been most involved in the imperialist plunder of other nations, so that would be the US, Canada, UK, France, Belgium, Germany, etcetera

The Imperial Core was clearly on Coruscant.

You know how any time there's a map where it colors countries who vote on UN resolutions, or countries where you can be thrown in jail for being poor, etc etc etc etc, you know how its usually a very similar map with US Europe and western allies on one side, and the entire rest of the world (the other 6.5 billion people) on the other?

Yeah, that teeny group that seems to always get its way controlling global politics is the imperial core.

Where is this imperial core you keep mentioning?

They prefer to be called The International Community™