The border checks will happen between Northern Ireland and Great Britain instead of between Ireland and Northern Ireland. In return for this concession, Northern Ireland will continue following some EU rules.
The country is officially known as The United Kingdom of Great Britain and Northern Ireland. Great Britain consists of 3 countries; England, Scotland and Wales. So the border Checks are between NI and GB not NI and UK.
Great Britain is the big island where England, Wales, and Scotland mostly are. There will be "border checks" down the Irish Sea, on goods which are travelling within the United Kingdom (specifically between Northern Ireland and Great Britain).
