EEA citizens are entitled to work legally in the Republic of Ireland and Northern Ireland. Non-EEA citizens with an Irish parent or grandparent are eligible for dual citizenship (and the right to work), although this procedure can be quite lengthy – enquire at an Irish embassy or consulate in your own country.
Full-time US students aged 18 and over can get a four-month work permit for Ireland, plus insurance and support information, through Work & Travel Ireland.
Most Commonwealth citizens with a UK-born parent are entitled to work in the North (and the rest of the UK) through the 'Right of Abode'. Most Commonwealth citizens under 31 are eligible for a Working Holidaymaker Visa: valid for two years, it allows you to work for a total of 12 months and must be obtained in advance. Check with the UK Border Agency (www.gov.uk/government/organisations/uk-visas-and-immigration) for more info.