Initially, you will be using a prepaid card when you come to Ireland. In order to have post paid plan you need to show Irish bank statements at least 3 months back and the usual references.  

The following mobile phone operators are providing services in Ireland:

Vodafone
www.vodafone.ie

Prepaid SIM card costs: €9.99, you can talk €15 value after registration
Pros:
Very good coverage, roaming is possible in most European countries with a prepaid card, Vodafone passport enables you to get excellent rates when roaming
Cons:
More expensive than the others

O2
www.o2.ie

Prepaid SIM card costs: €9.99, you can talk €15 value after registration
Pros:
Very good coverage, roaming is possible in most European countries with a prepaid card
Cons:
2nd most expensive

Meteor
www.meteor.ie

Prepaid SIM card costs: FREE (However, you have to top up with €10 at first usage.)
€50 call value after registration
Pros:
Good online service, the cheapest provider, cheaper international calls
Cons:
Poor coverage, roaming is unnecessarily overcomplicated you have to call them in advance and report the dates of you travel, to activate the roaming for you in a specific country.

There is a new mobile phone operator available recently on the market. It is not rated yet. You can find out more by visiting: www.three.ie