Can the European Council instruct the Commission?

Question

How is the relationship between the European Council and the Commission defined?

Can the Council instruct the Commission?

Answer 1

First a reminder of the confusing names:

European Council: The meetings of the Heads of Government
Council of the EU: The meetings of the governments of the various states, each represented by a minister. Previously called the "Council of Minsters".
Council of Europe: Not an EU institution.

The European Council sets policy, priorities and strategy, but it is not a legislative body. It doesn't create new European Law. On each matter they can either direct the Commission to create a proposal, which will then be considered by the Council of the EU, or direct the Council of the EU to deal with a matter directly.

So the European Council could decide "We want to reform the fisheries policy to better protect Mediterranian fish stocks" (policy) and ask the Commission to propose legislation to effect that change. The legislation would then be voted on in Parliament and in the Council of the EU.

Or the European Council could decide "We want to create a coordinated European Health Service" (strategy) and direct this matter to be dealt with by the Council of the EU (who would then direct the Commission to create proposals for this complex policy).

Formally the European Council makes requests, as the Commission is an independent body. However, a request from the European Council cannot just be ignored. The Commission has a duty to propose legislation in the interest of the EU. If they didn't produce a proposal they would be failing in this duty, and could be removed by a vote in Parliament. A request from "the boss" is de facto an instruction.

As the European Council is not a legislative body, it proceeds on consensus rather than on votes.