The European Commission is a council that is independent from all national governments in Europe. The word Commission refers to the members appointed by each country in Europe to manage the affairs of the EU. One member is appointed from each country to the EU. The word Commission is taken generally to refer to both the appointed members from each country as well as to the staff that manages the Commission.
The appointed representatives from each country are called the Commissioners. All of them are appointed from former elected officials in their own countries. Their responsibilitites while acting as Commissioners is to consider the good of the EU as a separate governing body rather than to owe allegiance to their native country. These Commissioners serve for five years when new appointments are made. Once appointed the Commissioners choose who to act as President of the EU. The current EU Commission President is Jose Manuel Barroso.
The Commissioners of the EU are appointed by the European Parliament. The European Parliament is an elected body of representatives. Once the Commissioners choose a President, he must be approved by the full body of the European Parliament. Once he is approved by the European Parliament, he picks the other Commissioners. The Commissioners are not elected like the other members of the European Parliament, they are picked by the President of the EU who has been chosen by the preceeding members and approved by the Parliament. One the approved President of the EU is approved and he has chosen his Commissioners, each one must be voted upon by the European Parliament.