British born citizens when traveling to Ireland do not require passports, but they should carry identification.  For citizens of the EU, Norway and Switzerland, the national identity card is enough (no passport required).  All visitors from other countries must carry a valid passport.

Citizens of some countries require a visa for entry into Ireland. Visas issued for the United Kingdom will not give you access to Ireland, you will have to apply for a seperate visa. Visas can be applied for at Irish Embassies and consulates abroad. For more details check with the Department of Foreign Affairs

There is no discernible border between the Republic of Ireland and Northern Ireland so normally no passport is needed when driving from one country into the other, if however you require a visa for either Ireland or the UK you must carry your passport with you when crossing the border.

Be aware, however, that once you cross the "border" into Northern Ireland speed limits and distances will be shown in miles (kilometres in the Republic). Northern Ireland uses pounds sterling, the currency in the Republic is Euro.