Yes, the European Union has a President: The President of the European Commision.
They are nominated for a five years period and are elected by the members of the European Council and approved by the European Parliament.
His duty is to control the Commission's Policy Agenda and the laws it produces. He also represents the EU abroad along with the President of the Council and the High Representative.
Since the year 2008 and until 2013 the President of the Europen Commision is Jose Manuel Barroso (Portugal).