How Will Ireland and associated Irish passport be affected by Brexit?
Ireland (commonly known as The Republic of Ireland, though the official name of the state is simply Ireland) is an independent country and an EU state. This status will not change. Ireland will remain a member state of the EU; Ireland issues its own passports and will continue to do so. Irish passports are EU passports.
Northern Ireland is part of the United Kingdom (full name: United Kingdom of Great Britain and Northern Ireland), and will leave the EU along with the rest of the UK. The constituent countries of the UK do not issue their own passports, so there is no such thing as an NI passport. UK passports will cease to be EU passports, even if they still have the words European Union written on the front cover. New UK passports will no longer have that text (and will be blue).