Healthcare is free to all residents of Ireland. However, most people have private health insurance with either VHI or BUPA; otherwise you could be in for a long wait if you have to go to hospital. 

Visitors from the United Kingdom to Ireland receive free urgent medical treatment automatically, as long as they can prove their citizenship or residence in the UK.  Visitors from EU/EEA countries are also entitled to free urgent medical care so long as they present form E111 which can be obtained from their own health services before visiting Ireland. 
However, here are exceptions to E111 cover. If you intend to stay abroad for some time, or are moving to Ireland to live permanently, then a separate form, E112, or local health insurance cover are more appropriate. The E112 can also be obtained from your local health board, which will explain to you how to complete it and what treatment is covered.

Working abroad

If you are going to work in Ireland for an employer from your native country or as a self-employed person, an E111 form might suffice, but check with your employer about what medical insurance they might provide for employees. If you intend to work in the Ireland for a foreign employer, then you need the E112.

Ireland also has a reciprocal health agreement with Australia, but visitors will be required to contribute a small amount towards their care. These agreements only apply if the visitor seeks medical attention from a public hospital or a doctor registered with the GMS.

All visitors from other countries are advised to take out a travel insurance policy that covers medical bills before visiting Ireland.

The Department of Health's website doesn't give much basic information, but has contact details for more questions.  For local health services information, check with your local healthboard.

http://www.irishhealth.com is a useful independent site.