EU countries do not have a single travel authority. 23 of the 27 EU members, four other large countries (Norway, Iceland, CH and Liechtenstein), and three other small countries (Monaco, San Marino and Vatican City) form the Schengen zone, whose travel rules are set by the EU. Three EU members - Bulgaria, Cyprus and Romania - have not joined the Schengen system yet. One EU member - Ireland - has its own system and does not even plan to join. Instead, it has open borders with Northern Ireland, which is part of the UK and therefore outside the EU. Hope this helps.

Edit: Ireland cannot join Schengen unless the UK also joins (because they are legally required to have open borders with NI), and the UK has no interest in joining.