Ulaidh – Cúige Uladh – Ulster
Ulaidh is the nominative, Uladh is the genitive. After simple prepositions, dative can be used: in Ultaibh, le hUltaibh.
Counties:
Aontroim – Antrim (gen. Aontroma)
Ard Mhacha – Armagh
an Cabhán – Cavan (gen. an Chabháin)
Doire – Derry
an Dún – Down (gen. an Dúin)
Dún na nGall – Donegal (Tír Chonaill – Tyrconnell)
Fear Manach – Fermanagh
Muineachán – Monaghan (gen. Mhuineacháin)
Tír Eoghain – Tyrone
Laighin – Cúige Laighean – Leinster
Laighin is the nominative, Laighean is the genitive. After simple prepostions, you might see the dative form: i Laighnibh.
Counties:
Áth Cliath – Dublin
Ceatharlach – Carlow (You’d expect this -ach to become -aigh or -aí in the genitive, but the recommended genitive form is Cheatharlach, i.e. with just the first consonant mutated.)
Cill Chainnigh – Kilkenny
Cill Dara – Kildare
Cill Mhantáin – Wicklow
an Iarmhí – Westmeath (gen. na hIarmhí)
Laois (gen. Laoise)
Loch Garman – Wexford. Actually, the form is etymologically speaking Loch gCarman, with an eclipsis that isn’t a regular feature of the language anymore (similar fossilized forms are found in an Muileann gCearr ‘Mullingar’, Loch nEathach ‘Lough Neagh’ etc.); Wexfordmen are called Carmanaigh in Irish.
an Longfort – Longford (gen. an Longfoirt; it would make more etymological sense to write it as an Longphort)
Lú – Louth
an Mhí – Meath (gen. na Mí)
Uíbh Fhailí – Offaly
An Mhumhain – Cúige Mumhan (!) – Munster
an Mhumhain, gen. na Mumhan.
Counties:
Ciarraí – Kerry
an Clár – Clare (gen. an Chláir)
Corcaigh – Cork (gen. Chorcaí)
Luimneach – Limerick (gen. Luimnigh)
Port Láirge – Waterford
Tiobraid Árann – Tipperary
Connachta – Cúige Chonnacht – Connacht
Connachta, gen. Chonnacht, dative Connachtaibh (i gConnachtaibh)
Counties:
Gaillimh – Galway (genitive: na Gaillimhe; in Munster Irish, “in Galway” is “sa Ghaillimh” rather than “i nGaillimh“)
Liatroim – Leitrim (genitive: Liatroma)
Maigh Eo – Mayo
Ros Comáin – Roscommon
Sligeach – Sligo (genitive: Shligigh)