Which type of government system best describes the relationship between member nations in the European Union?

The correct answer is - federal government system.

Even though all of the countries in the European Union are separate countries and they all have their own governments, still the European Union has authority over them. The European Union has its own parliament, and one of the main conditions for every country member is that it obliges to respect and implement the decisions of this parliament, thus losing little of their sovereignty in the process. This is pretty much the same principle used in the federal government systems, where all states have their own leaders and have big authority in their own borders, but also have to obey the commands and decisions of the central government.