Ireland is a safe place and this is due to the the Irish being a friendly gentle people. There are two law enforcement agencies on the island.
In the Republic of Ireland the law enforcement agency is called An Garda Siochana (Gardai). In the North the police force is known as the Police Force of Northern Ireland (PSNI). The Guards as they are called are very helpful if you are in need of any help or just lost and looking for directions.
Like any other country, when traveling keep your money in a safe place and take the usual precautions when walking alone on streets by night. Dublin is like all big cities and so take all the usual precautions. If you have any emergency while in Ireland dial 999 or 112 toll free from any phone and ask for the service you require.
In the North of Ireland during "The Troubles" there was never any significant risks to tourists and with the present ceasefire almost continually in operation since August 31st 1994 there never has been a better time to visit this beautiful part of the country.
In the North of Ireland all emergency services (Police, Fire, Ambulance and Coastguard Services) can be reached by dialing 999 toll free from any telephone.