Well, the previous posters have given plenty of good ideas, so I will just list a couple of historical sites in chronological rather than geographical order, and off the top of my head. Do bear in mind that I'm not a historian, so this may be tendentious or inaccurate in places... Dublin obviously has numerous places of historic interest from the eleventh century on, while Belfast is of particular interest due to the troubles of the later 20th century (and its earlier role as the country's main industrial centre).
Tara, Co. Meath, semi-mythical capital of the High Kings of Ireland; other nearby sites include the prehistoric tombs at Newgrange and the site of the Battle of the Boyne in 1690 in the war between the Protestant-backed William of Orange, the Catholic-backed James II of England, and Louis of France. Aughrim was the really decisive battle, I think, but the Boyne is the one which is most remembered and commemorated. On the Hill of Slane, also nearby, St Patrick is said to have lit the Paschal fire in 433 AD in defiance of the High King who could see it from Tara.
Cashel, Co. Tipperary - historic seat of the Eoghanacht dynasty and later ecclesiastical capital, with a number of interesting religious monuments.
Grianán of Aileach, Co. Donegal - seat of the Uí Néill dynasty.
Clonmacnoise, Co. Offaly - major monastic site dating to the 6th century.
Kilkenny - first seat of the parliament of the English colony in Ireland, if I'm not mistaken. Kilkenny was also the seat of the Confederation which brought together the native Irish and Anglo-Normans in opposition to the Puritans during the English Civil War.
Trim, Co. Meath - one of the more impressive remaining castles from the Norman period.
Kinsale: the Battle of Kinsale in 1601, when the forces of the Irish princes O'Neill and O'Donnell marched from their northern bases but failed in their attempt to relieve the Spanish force that had landed in the town, effectively spelled the end of Gaelic Ireland and the completion of the English conquest.
Lough Swilly, Co. Donegal: the Flight of the Earls (O'Neill and O'Donnell) from Lough Swilly to continental Europe in 1607 sealed that process.
Aughrim, Co. Galway - field of the decisive battle in the war between William of Orange and James II; the victory of the former ended tolerance of Catholicism and established the Protestant ascendancy in Ireland.
Londonderry/Derry - Protestant-held city besieged by Jacobite forces in 1689, the failure of the siege is still commemorated in Northern Ireland. The old city walls are still in place, and it's quite a nice town for a visit.
Limerick: here the Jacobite forces capitulated in 1691 having signed the Treaty of Limerick with the Williamites. The Treaty Stone on which it is supposed to have been signed is one of the city's best-known monuments.
Vinegar Hill, Co. Wexford: here the rebels of the 1798 uprising made their final stand. There were other battles at Wexford, New Ross and other places in Wicklow, Castlebar in Co. Mayo, and various places in Co. Antrim, notably Ballinahinch. The last rebels, under Michael Dwyer, held out in the Wicklow mountains for several years after 1798. The road between Killakee and Glendalough via Sally Gap, still called the Military Road, was built by the British Army to allow it access to the mountains, and one of the military barracks built along it, at Glencree, is now a peace and reconciliation centre which runs a variety of programmes and has occasional exhibitions as well, with a cafe that's open to the public. There is a German military cemetery across the road from the centre. The Military Road is also extremely scenic. Glendalough itself is an interesting and attractive monastic site dating to the 6th century.
Doolough or Delphi, south of Louisburgh, Co. Mayo - site of the death of numerous victims of the Great Famine in 1849. If I remember the details, the unfortunate victims had walked from Louisburgh to seek assistance from the district relief commissioners, who were meeting at Delphi Lodge, and after they were refused help many perished on the road back. It's a beautiful and desolate place. A memorial walk takes place there every year now.
Obviously, there's no direct correlation between the historic significance of what happened somewhere and how interesting it is to visit now - the battlefields of Aughrim and the Boyne (especially the former) would require some imagination to conjure up images of the historic events that took place there.