A very quick description of who does what in the EU. (Since the EU is currently divided into 3 areas, or "Pillars" and not every institution has the same amount of power in each, the question of "who does what" can get confusing...):
The European Commission: A cross between a government and a civil service. This is made up of unelected Commissioners, with one appointed by each member state every five years under the current system. It can propose laws but it can't pass them, and this power is also limited to certain areas and certain circumstances (mostly Pillar 1, which is mainly about the single market and other technical issues). It is supposed to enact EU legislation and/or make sure that those who should be doing this, are. So if you read that "the Commission has decided...", it very rarely gets to decide anything: its main function is suggesting and drafting laws.
The European Council: This is made up of the heads of government (who are also heads of state in some cases), and they meet at least twice a year. This body decides the big political issues. The presidency of this body is taken on by each of the member states in turn for 6 month long presidencies under the current system.
The Council of the European Union: the main legislator. Made up of national ministers from whatever area the issue in question involves (agriculture ministers if the issue is the CAP, etc), although the vast majority of the work is done by national civil servants. NO major legislation can be passed without this body's say-so. Can ask the Commission to draw up legislation. The main institution in all legislative areas.
The European Parliament: made up of directly elected MEPs. In most of Pillar 1 they have an equal say on legislation (amending, rejecting or passing it) as the Council of the European Union, but in Pillars 2 & 3 (foreign policy and home affairs) they have no real say. Can ask the Commission to draw up legislation under Pillar 1.
The European Court of Justice: a Court which decides on issues of EU (or "Community") law. This includes constitutional matters (which institution has the right to do what), and interpreting Community law for national courts. Has little to no role under Pillars 2 & 3, however.
The Court of Auditors: the accountants. They check and investigate the finances of the EU and its institutions.
Note: the Council of Europe has nothing to do with the EU. It was a body born out of the federalist movement, but it is an intergovernmental body and is most famous for the European Convention on Human Rights and the European Court of Human Rights in Strasbourg.