Appointing the President
The candidate is put forward by national leaders in the European Council taking account of the results of the European Parliament elections. He or she needs the support of a majority of members of the European Parliament (the 28 heads of government of the EU member states that includes the UK) in order to be elected.  

Commissioners
The Presidential candidate selects potential Vice-Presidents and Commissioners based on suggestions from  all the EU countries (including the UK). The list of nominees has to be approved by national leaders in the European Council.

Each nominee appears before the European Parliament to explain their vision and answer questions. Parliament then votes on whether to accept the nominees as a team. Finally, they are appointed by the European Council, by a qualified majority vote.

The current Commission’s term of office runs until 31 October 2019.

The EU Commission can only PROPOSE laws where EU governments have unanimously agreed to allow under the EU treaty.   The Commission can only propose laws in areas where the UK and the House of Commons  allows it to do so, and ‘PROPOSING’ is not ‘DECIDING’ laws; laws that only become law when approved by both a qualified-majority in the EU council and a simple majority in the European Parliament.

The EU isn’t perfect.   Having existed for a relatively short period of time, it allowed for  a more democratic process;  the main European parties proposed rival candidates for the Commission President before the 2014 European Parliament elections.  After the European People’s Party (EPP) won the most seats in the new Parliament, the European Council agreed to propose the EPP’s candidate: Jean-Claude Junker.

However, the UK Conservatives left the EPP in 2009 leaving UK voters unable to vote for Junker, but they could vote against him.   The British therefore cannot fault the EU for the Conservatives leaving the EU.

So, the President  having been chosen, EVERY EU member state (including the UK) nominates a Commissioner.   Each  Commissioner then has a hearing in one of the committees of the EU Parliament.  If a committee issues a negative opinion, the candidate is usually withdrawn by the government concerned.   After the hearings, the team of 28  is subject to in investiture vote by a simple majority of the MEPs. 

Finally, once invested, the Commission as a whole can be removed by a two-thirds censure vote in the European Parliament.  Although this has never happened , in 1999 the Santer Commission resigned before a censure vote was due to be taken because they were likely to lose.    So it is not strictly true to say the Commission is unelected or indeed unaccountable.

 Similarities can be found in the UK Parliament that even though it has existed for centuries, still retains an unelected House of Lords that can scrutinise legislation and Government policy.   The leader of a particular UK party is unelected but rather chosen by members of their party, who is then invited by the Queen (ceremonial) to become Prime Minister who then names his or her own cabinet and ministers.

Again, the EU in its short existence is not prefect, it will continue to ensure a more democratic process, but a post-Brexit UK will not be involved.