From the EU's own website:
The Commission is the European institution that has the monopoly on legislative initiative and important executive powers in policies such as competition and external trade. It is the principal executive body of the European Union and it is formed by a College of members composed of one Commissioner per Member State. It also chairs the committees responsible for the implementation of EU law. The former comitology system has recently been replaced by new legal instruments, implementing and delegated acts.
Here in the UK, people often have a slightly confused view of government and parliament. This is because our government is made up of elected MP's chosen from parliament by our Prime Minister who is also an elected MP,.
We don't really see government/parliament as that different we focus more on the 'party of government' and 'party of opposition'.
Compare this to the USA where they have two houses (The Senate - like an elected house of lords, and The House of Representatives - like our elected 'backbench' MPs) but their government (the executive) is the separately, directly elected President and anyone who he appoints as his 'ministers', none of whom have to have been elected or ever have been a Senator or Representative.
The EU has the worst of both of these worlds - unlike the UK the EU government/executive is not made up of elected MEPs, it is appointed by unelected bureaucrats - but further even the president who runs it all isn't elected either!
The EU president may as well be like the English King John who ruled as absolute monarch until he was put under the 'Magna Carta' which established in law the rights of the English to be ruled by an elected parliament to whom even the King was answerable.
The EU government strips away 800 years of the best of English liberty and democracy. It is horrific that this has been allowed to happen - and even more horrific that people are arguing to let it continue by voting to remain in the EU at the upcoming referendum.