The European Parliament may approve or reject a legislative proposal, or propose amendments to it. The Council is not legally obliged to take account of Parliament’s opinion but in line with the case-law of the Court of Justice, it must not take a decision without having received it.
http://www.europarl.europa.eu/about-parliament/en/powers-and...
The European Commission drafts legislation and then the European Parliament decides whether or not to amend or pass the legislation.
Hence, this appointment system, which is additionally kept in check by the European Parliament whose members are elected directly by the voters from all member states.
You could complain that it's too representative, but not that it's not democratic. The alternative of giving the European Parliament more power and let it constitute a "European Government" directly has not been found appealing by the governments of its member states, particularly not by the UK, since they do not want to give away so much of their sovereignty.
Complaining about lack of democracy in the EU while at the same time complaining about lack of national sovereignty is perhaps the most hypocritical and intentionally misleading part of the current populist agenda.
Oh I know you'll tell me of the ignorance and stupidity of the racist, little England British voter and how they need to be 're-educated' but this is hardly a mandate from the people is it?
> ignorance
People can't be proud of not knowing about the EU and complain about being called ignorant at the same time.
(Quite a lot of this is misinformation, frankly - the Mail and the Express run hilariously biased anti-EU campaigns on their front pages.)
Our media are at least partly to blame, tolerating "opinions" on matters of fact and minimising our real exposure to actual things the EU does.