EU is a strange creature.
Basicaly the idea is based on a free flow of capital, goods & people among the members (and also some political ones: eg. some common foreign policy, agricultural policy, interior affairs (immigration) policy etc). But ... There are several sub-systems. Eg. EU has now 27 members. Only 13 among them are in Eurozone (using single currency: Euro; others haven't met criteria for adopting the Euro, some ie. UK, Denmark & Sweden refused to use Euro). 13 of EU members (+ Non-EU Iceland & Norway) are in "Schengen zone" which allows completly free flow of people (no border control between Schengen members). But those 13 are not the same as the one in Eurozone (eg. Sweden & Slovenia are switched). Also there is no free labour market since some of the old EU members (15 of them) have applied some restrictions to labour force from some new EU members. Also there are some other common policies (eg. agricultural etc) but with many exclusions.
So, EU is not a homogenous union, but there are men sub-unions (eg. Eurozone, Schengen zone, free labour market etc).
Regarding sovereignty: each EU members should comply to some common EU rules & standards, but some have gained some exceptions during the association negotiations. All members are obliged to adopt Euro & enter Eurozone when met all the convergence criteria (except UK & Denmark - who have a right to refuse entering Eurozone). Also all EU members are obliged to enter Schengen zone when meeting the security criteria.
Eurozone have common monetary policy (eg. setting common interest rates), while fiscal policy (public spending & tax policy) is still in a domain of each country. But every country is obliged to stay within convergence criteria (low inflation, low budged deficit etc). Also every EU member is obliged to pay some of its GDP to common EU budget which then redistributes the money to poorer parts of EU (you get a "free" money for a specific project). So, rich EU members are net givers (giving more than receiving back), while poorer ones are net receivers (receiving back more than giving into EU budget).
It's complicated.
Btw, use Wikipedia to find more interesting info about EU, Eurozone, Schengen etc.
