Ó Conchobhair of Connacht
Though not descended in the male-line from them, the current Royal House is technically a part of the Ó Conchobhair dynasty of the Kingdom of Connacht. There are several branches of this dynasty and many titles attributed to it. The current chief of this clan is Desmond Roderic O'Conor, The O'Conor Don, who is the rightful King of Connacht, but there are many more branches, some of which still exist.
Ó Conchubhair Donn
The Ó Conchubhair Donn (anglicised: O'Conor Don) branch of the O'Connor dynasty of Connacht is the leading branch of the entire dynasty. The current head of this sept is Desmond Roderic O'Conor of Sussex, England. They previously were the owners of Clonalis House in Co. Roscommon, Ireland, but currently the owner is Desmond's cousin Pyers O'Conor Nash, the grandson of a previous Chief of the Name.
The O'Conor Don is Chief of the Name of the entire O'Connor dynasty. He was recognised as such in the 1940's by Chief Herald of Ireland Edward MacLysaght.
Desmond O'Conor Don has been The O'Conor Don since the death of his father Denis Armar O'Conor Don in 2000.
The O'Conor Don descends in the male-line from Feidhlim Geancach Ó Conchubhair Donn, the last King of Connacht. Therefore, whoever is recognised as O'Conor Don is also recognised as the rightful King of Connacht.
The O'Conor Don line descends in the male-line from Cathal Crobderg Ua Conchobair, King of Connacht, youngest son of Tairrdelbach Ua Conchobair, High King of Ireland, and King of Connacht.
Clonalis House has its own website at https://clonalishouse.com.
Ó Conchobhair Ruadh
The Ó Conchobhair Ruadh (anglicised: O'Connor Roe) branch of the O'Connor dynasty is closely related to the O'Conor Don branch. They have a common agnatic ancestor in Aedh Ó Conchobair, King of Connacht (died 1309), who himself was an agnatic descendant of King Cathal Crobderg of Connacht. The man who most likely would be head of this sept is Douglas R. O'Connor of Oregon, U.S.A.
Douglas O'Connor Roe, although not recognised officially in this title, has been the senior claimant to it since the death of his father, Frederick Drexel O'Connor, in 2004.
They provided one King of Connacht: Fedlim Ó Conchobair Ruadh (died 1316). Beyond that they were lords of Kilgefin.
Ó Conchobhair Sligigh
The Ó Conchobhair Sligigh (anglicised: O'Connor Sligo) branch of the O'Connor dynasty were a less prominent branch. They descend agnatically from Brian Luighnech Ua Conchobair, the elder half-brother of Cathal Crobderg. The current head of this branch would almost certainly be John Patrick O'Connor III of Illinois, U.S.A.
John Patrick O'Connor Sligo III, though not officially recognised, has been senior known claimant to the title since the death of his father John Patrick O'Connor II on 12 November, 2015.
This branch of the dyansty also only provided one King of Connacht: Cathal mac Domhnaill Ua Conchobair Sligach (died 1324). The current claimant and most subsequent O'Connor Sligos descend from Cathal mac Domhnaill's brother, Lord Muirchertach of Cairpre.
This family held the title Lord of Sligo until the death of Sir Donnchadh Ua Conchobhair Sligach circa 1634. His first-cousin Martin was the ancestor to the current heir.
Ó Conchobhair Bréifnech
The Ó Conchobhair Bréifnech (anglicised: O'Connor Breifne) branch of the O'Connor dynasty used to be one of the most prominent. Also called Clann Muircheartaigh Uí Conchobair (anglicised: Clan Murtagh O'Connor), they were the agnatic descendants of Muirchertach Muimhnech Ua Conchobair, a half-brother of Brian Luighnech and Cathal Crobderg. There is currently no identifiable head of this branch, and they may have actually died out (though it should be noted that His Royal Highness Prince Díolún believes the O'Connors of Swinford to be related). The last known head of this sept was Séaghan Ó Conchobhair Bréifneach, also known as Shane/John Connors, who died around 1720, and descended from Lord Donnchad mac Muircheartaigh Ó Conchobhair Bréifnech, who died in 1474.
This branch of the dynasty actually provided several Kings of Connacht (four, to be precise), and was the second-to-last branch to do so; the last branch to do so was the O'Conor Dons. The last King of Connacht from this branch was Aedh Ó Conchobhair Bréifnech (died 1350).
This family, in addition to King of Connacht, held the titles Lord of Iorruis (Modern Irish: Iorrais; English: Erris), and Lord of Baile an Róba (anglicised: Ballinrobe). They also held considerable lands in the Mayo Abbey area.
Mac Maghnusa of Kilronan
The Mac Maghnusa (anglicised: MacManus/McManus) branch of the O'Connor dynasty is considerably obscure, and much lesser-known than any of the others. In fact, it may not be outlandish to say that most do not know that they are a branch of the O'Connor dynasty. They are agnatic descendants of Maghnus Ua Conchobair, a half-brother of Brian Luighnech and Muirchertach Muimhnech, and a full-brother of Cathal Crobderg (the other being Brian Bréifnech Ua Conchobair).
Currently, the MacManus branch has no recognised or identified chief, or heir to such a title. Anyone the Imperator Scottorum has come up with has been pure conjecture. It cannot even be assumed that a person with this name or a variant is from this branch of the dynasty, as there is a completely different Irish clan from Tír Chonaill (now Donegal) called Mág/Mac Maghnais (anglicised as MacManus/McManus).
This branch provided no kings of Connacht. They were, however, lords of Tír Tuathail (now called Kilronan) in Co. Roscommon. The last to hold this title were two brothers jointly, Toirdhelbhach Buidhe Mac Maghnusa and Brian Mac Maghnusa, who were executed by English forces on 3 March, 1586.
House of Ó Conchobhair
The House of Ó Conchobhair (anglicised: House of O'Connor), also called the Síol Ruaidhrí or Clann Ruaidhrí Uí Conchobhair were the direct, male-line descendants of Ruaidrí Ua Conchobair, the last High King of Ireland and the only King of Ireland. Ruaidrí was the half-brother of Brian Luighnech, Muirchertach Muimhnech, Maghnus, and Cathal Crobderg. They were all sons of Tairrdelbach Ua Conchobair, High King of Ireland.
There were three main branches, and they descended from his three sons who became Kings of Connacht: Conchobar Máenmaige, Toirdelbach Mór, and Aedh mac Ruaidrí.
The House of O'Connor was a branch of the Ó Conchobhair dynasty of Connacht and they are not two names for the same thing.
The last head of the House of O'Connor was Richard O'Connor (Ruaidhrí Ó Conchobhair) who died in 1904. With his death, so too did the House of O'Connor die out.
The Mac Bréifnech branch of the Ó Conchobhair dynasty is even more obscure than the Mac Maghnusa. They are the only living descendants of Ruaidrí Ua Conchobair, the last High King of Ireland and the only King of Ireland. Ruaidrí was the half-brother of Brian Luighnech, Muirchertach Muimhnech, Maghnus, and Cathal Crobderg. They were all sons of Tairrdelbach Ua Conchobair, High King of Ireland.
The Mac Bréifneach family are named for Ruaidhrí Bréifneach Ó Conchobhair, elder brother to their ancestor Conchobar.
The current head of the house is Prince Dáibhid, Duke of Munster. He is the grandfather of the current claimant to the Irish High Kingship, Prince Díolún, Duke of Meath. Dáibhid became such upon the death of his father, Fionnladh, in July, 2014.
The Mac Bréifneach and their descendants are all that remain of the House of O'Connor and Síol Ruaidhrí.