Ireland (Republic) does NOT use the punt any longer. It uses the Euro and has done for years.
The good news is you don't need to change money if going from continental Europe to Ireland (or vice versa).
Northern Ireland uses the pound sterling (as used in Scotland, England and Wales) and issues its own banknotes too (which are in pound sterling but are not widely recognised or accepted in England despite being legal tender).
You will, therefore, have to deal in 2 currencies. On the Irish border it is common for shops to accept both Euros and pounds.
If in Northern Ireland, make sure to spend your Irish bank sterling notes in Northern Ireland (e.g. Northern Bank, Bank of Ulster), and ask for English notes when you are getting change. Notes come in the standard denominations of 5, 10, 20 and 50 pounds.
The coins are the same all over Northern Ireland and the rest of the UK. Euro notes and coins are not accepted in England, Scotland or Wales (but notes are easily changeable in UK post offices, travel agencies or banks).
Hope this helps you