A defining principle that determines the level where policy action is to be taken in the European Union is the concept of subsidiarity.
Read this small passage on subsidiarity
So to summarise, the subsidiarity principle is the idea that political action should be taken as close to the citizen as possible. This means that in an instance where the EU and a member state are both able to take action, action should ordinarily be taken by the member state.
Caveats are included that mean the EU will take action if the action to be taken is within the sole jurisdiction of the EU (member states are not supposed to act) or of the member state is unable to sufficiently undertake the action.
Support for subsidiarity is complicated. States and organisations that are pro-integration usually support subsidiarity on the basis that it is federalist in nature. Those states that tend to be reticent towards further integration (such as some UK governments) approve of subsidiarity on the basis that it protects the power of national government.
The EU’s record on subsidiarity often comes under criticism. Read this highly critical news article from the Financial Times regarding the lack of application of subsidiarity. (You may need to register, for free, on the FT website to view the article).