The European Commission is the executive of the EU, these are the people who run the EU. It is made up of one commissioner for each member state, so currently there are 27 of them. Each one is bound by an oath to represent the best interests of the EU as a whole, not the best interests of their own home member state.
They propose new legislation (and are the only people who can do this), Implement decisions, uphold the treatys and look after the EU on a day to day basis. The term commission can apply to the 27 commissioners (collectively known as the College) or it can also apply to the much larger institution (about 25,000 European civil servants)
There is a president of the EU commission (one of the 27 mentioned above) who is appointed by the EU council first. There are criteria as to who the council can choose, they are preferably from southern Europe , they must be fluent in the French language, there home state must also be a member of the euro zone and the schengen afgreement. Then the president must be elected by the parliament, this is done by a simple yes/no vote, in the case of a no vote (which has never happened) the council will select another president. Then the council and the EU commission president together appoint the other 26 members.
At this point the 27 commissioners as a whole must be approved by the EU parliament. The parliament can only decide yes or no to the commission as a whole, it can not amend choices or propose others and can not vote on individual members.