Dear God, Northern Ireland is part of the United Kingdom... look at the front of your passport. You don't even need a passport to go the the south of Ireland. Enjoy the visit.
Ireland is divided in two. The South which is independent and part of the North which is part of the United Kingdom i.e. British. Both parts are in the Schengen Agreement area. Have a look at http://en.wikipedia.org/wiki/Schengen_Agreement for more information.