The Court of Justice of the European Union – to give it its full name – is the EU’s highest legal authority. It is based in Luxembourg. It is an entirely different thing to the European Court of Human Rights (ECHR).
It is the European Court of Human Rights, not the ECJ that has often upset British politicians by making it harder, for example, to deport terrorist suspects. The ECJ interprets and enforces the rules of the single market, settling disputes between member countries over issues like free movement and trade. It is at the center of pretty much everything the EU does and it having the power over UK actions has been a key issue for those arguing for the UK to leave to the EU to regain full sovereignty.
Prime Minister Theresa May has vowed that Britain will not be under the “direct” jurisdiction of the ECJ after Brexit. But she has suggested that elements of relations could – where the UK signs up to specific EU agencies – still be covered by the ECJ after Brexit
After that, there will need to be a new mechanism for settling disputes between the UK and the EU but what form that take has yet to be decided. There has been talk of an ombudsman, or some other third party, being appointed to settle disagreements.
The version of the Brexit deal, published on 8 December 2017, do also give limited powers to the ECJ in terms of EU citizens living in the UK for up to eight years. The political declaration document makes clear that ECJ will continue to have a role on interpreting EU law after Brexit.