Damn. I wish there was more Yes, Minister...
Having three different European bodies referred to as councils is a bit silly. I wish they'd change the names. They probably will eventually.
The explanation is rather Humphreyic though. It can be made much easier:
Europe -- meaning the European Union -- is run by the sovereign governments of the member states, and the people of the European Union.
The
European Council is really just the formal meeting ground for the heads of states/governments of the member states. Appoints the President of the Commission.
The
Council of the European Union acts like the legislative "upper house" of the EU system, and consists of one minister from each member state (which minister depends on the topics at hand).
The
European Parliament is the legislative "lower house" of the EU system. Members are elected by the people of the European Union in normal, free and open elections. Approves the European Commission and may dissolve it. Has a President of the European Parliament, which functions as parliament's speaker.
The
European Commission is the executive branch. After each EU Parliament election the European Council nominate the
President of the European Commission (same way as the Queen appointing the Prime Minister in the UK) which the Parliament then ratifies. An approved Commissioner, together with the European Council, then selects his cabinet (one member from each member state) and has to get them approved by the Parliament.
Not especially contrived or conspiratorial, is it?
Also, you did notice this part, I take it:
Sir Humphrey What did you want?
Minister Sovereignty.
Sir Humphrey We had that already. What else?
Minister Control of our borders.
Sir Humphrey We had that too. Anything else?