Inspiration cred goes to Professor Mårten Schultz, who recently made a list of mistakes commonly made when discussing and reporting on legal matters, and Annika Ström Melin's article about lacking EU knowledge, which itself was wrongly edited.
1. European council (Europeiska rådet)
The European council consists of the heads of state of the EU's Member States. The European Council does not have any formal legislative power! Instead, the European council sets out the broader political direction of the EU. Sometimes, one can get the impression that the European Council enters into international agreements, such as the recent deal with Turkey regarding asylum seekers. However, such agreements, if made by the European Council alone, are not necessarily legally binding. For more detail on this issue, read here.
2. Council of ministers (Ministerrådet) (also called the Council of the European Union)
This is the Council that is meant when just referring to "the Council". The council is made up of the ministers of the different Member State governments. The council has a number of different constellations, depending on the question discussed. For example, when discussing the regulation of foodstuffs (say bananas), the ministers responsible for that issue will meet. The Council, together with the European Parliament, has the formal legislative power in the EU.
3. European Parliament (Europaparlamentet)
The European Parliament is directly elected by the citizens of the EU and is not organized by nationality of its members, but by groups of political parties. Together with the Council, the Parliament has the legislative power in most, but not all, legislative issues decided by the EU.
4. Commission (Kommissionen)
The Commission consists of one Commissioner from each Member State. The Commissioners usually have a background as politicians and are (similar to ministers) tasked with different policy areas. One of the most important tasks of the Commission is to propose and draft new legislation. Thus, when you read in the paper that "the EU wants (insert controversial issue)", usually what has happened is that the Commission has proposed new legislation. This, however, does not necessarily mean that such legislation will also be passed, since the Commission does not have legislative power to adopt its proposals. This power rests with the Council of ministers (indirectly elected by the people) and the Parliament (directly elected by the people).
5. European Court of Justice / Court of Justice of the European Union (EU-domstolen)
The Court has its seat in Luxembourg and actually consists of three separate Courts: The Court of Justice, the General Court and the Civil Service Tribunal. Most prominently, the Court of Justice answers questions regarding the interpretation of EU law by national courts of the Member States (so called preliminary rulings). Note that preliminary rulings are just that - preliminary. Final judgment is always rendered by the national court asking the question.
II. Institutions NOT part of the EU
6. Council of Europe (Europarådet)
The Council of Europe, which has 47 European states as its' members, is an international organization tasked with promoting human rights, democracy and the rule of law in Europe. The monumental achievement of the Council of Europe is the adoption of the European Convention on Human Rights (ECHR) in 1950, Europe's human rights bill. The Council of Europe is also, contrary to the popular belief that the EU is the originator, the creator of the gold-starred European flag and the European anthem.
7. European Court of Human Rights (Europadomstolen)
This Court has its seat in Strasbourg and is tasked with judging in cases where one of the states signatory of the European Convention of Human Rights (ECHR) has allegedly infringed human rights.
III. EU Legislation and decision-making
8. Regulations (Förordningar)
Regulations are one of the legislative instruments that can be adopted by the EU (usually the Council and the Parliament together). Regulations are directly applicable in the Member States. One example it Regulation 1/2003 which, amongst other things, allows the Commission to investigate and fine companies that have been involved in a cartel.
9. Directives (Direktiv)
Directives are the other important legislative instrument that can be adopted by the EU. Directives need to be implemented into national law by the Member States. This means that a national law can be based on a directive, but this is not necessarily visible from the individual clauses in the law itself.
10. International free trade agreements (Internationella frihandelsavtal)
If the Member States want to enter into a free trade agreement with a state outside of the EU, they first need to give the Commission a mandate to negotiate such an agreement. During negotiations, the Commission must inform and consult with the Council and the European Parliament. Once negotiations are concluded, the Council and the European Parliament must formally agree to the deal negotiated by the Commission. Recent examples are the free trade agreements with Canada (CETA) and with the USA (TTIP), whereas the latter is still being negotiated.