We did not vote for them.
We do vote for and elect MEPs; and we don't vote for UK Cabinet ministers.
Result: Neither
On 22nd May 2014 we elected our MEPs - aka "Member of the European Parliament".
MEPs vote on who will be the President of the Commission (since November 2014 that has been Jean-Claude Juncker). This is similar to how our MPs select the Prime Minister.
Then there are 33,000 "unelected bureaucrats", aka the European Commission, who are very similar to the 400,000 UK civil servants - none of which are elected. They mainly propose new legislation (which the MEPs vote on), and work on the budget/funding of the EU (source).
There are 751 MEPs in the European Parliament, of which the UK send 73. This is based the size of the population, rather than an even split between the 28 countries, so we represent 9.7% of the MEPs (not 3.7%).
Cannot find original source, but more can be found on swindonforeurope.co.uk.
If you disagree, or would like to add more details, please let me know.
You can suggest a reason for leaving the EU, or look at the other reasons.