Your IRP card is your permission to stay in Ireland. As of 13 May 2019 it doubles as your re-entry permission. That means you do not require a reentry visa anymore. However, if you wish to travel outside of the Republic of Ireland you may require a visa.
It is your responsibility to check what the visa requirements are for the country you wish to travel to.
Northern Ireland & UK
Please note Northern Ireland is part of the UK. If you wish to visit Belfast, the Giant’s Causeway etc. you may require a UK visa. You can check here if you need a visa to travel to the UK.
If you wish to travel on the European continent you may require a visa, a so called Schengen visa, which allows you to travel to any member state of the Schengen area. You can check here if you require a Schengen visa.
To apply for a Schengen visa you will have to make an appointment at the embassy of the country you wish to enter the Schengen area from or the embassy of the country you intend to stay the longest in. For your appointment you will require a travel visa letter which you can request on NCI360.
You may require a visa to travel to any other country, e.g. United States or Canada. If you are not sure whether you require a visa you can easily check here