In reference to Irish geneology, what exactly is the Line of Heber?

This I can answer as my family is part of this line :)

LINE OF HEBER

Heber is the line from which Brian Boru comes, as do:
Brady, Brennan, Carroll, Casey, Clancy, Coghlan, Connell, Cullen, Doran,
Hagerty, Hickey, Hogan, Kearny, Kelleher, Kennedy, Lynch, Lyons, Lysaght, McCarthy,
MacGrath, MacMahon, Macnamara, Moroney, Moloney, O'Brien (10 different
pedigrees), O'Callaghan, Collins, O'Connor, O'Corcoran, Daly, Donoghue,
Donovan, Flanagan, O'Gara, O'Grady, O'Hara, O'Keeffe, Liddy, Mahony,
Meagher, O'Meara, O'Neill, O'Sullivan, Plunkett, Power (o'Poir), Quin, Quaile, Ring,
Shannon, Slattery, Stewart, Tracey - to name a few.
http://www.libraryireland.com/Pedigrees1/Heber.php

38. Conmaol; 12th Monarch c.1650 BC

39. Eochaidh Faobhar Glas; 17th Monarch c.1492 BC

40. Eanna Airgthach; 21st Monarch c.1409 BC

41. Glas

42. Ros

43. Rotheacta

44. Fearard

45. Cas

46. Munmoin; 25th Monarch c.1332 BC. Ordained his Nobles to wear gold necklaces.

47. Fualdergoid; 26th Monarch, c.1327 BC, ordered his Nobles to wear gold rings.

48. Cas Cedchaingnigh; Revised the study of the laws, poetry, and sciences
which had become little practised since the death of Amergin the Druid.

49. Failbhe Iolcorach; Ordered stone walls be built between neighbors'lands.

50. Ronnach


51. Rotheachta; 35th Monarch c.1030 BC

52. Eiliomh Ollfhionach

53. Art Imleach; 38th Monarch c.1013 BC

54. Breas Rioghacta; 40th Monarch c.961 BC

55. Seidnae Innaridh; 43rd Monarch, c.929 BC, and first to pay his soldiers and put them
under disipline. Previously their pay was what they could get from their enemies.

56. Duach Fionn; died B.C. 893

57. Eanna Dearg; 47th Monarch, c.892 BC, died suddenly, with most of his retinue,
adoring their gods at Sliabh Mis, B.C. 880.

58. Lughaidh Iardhonn

59. Eochaidh

60. Lughaidh; died B.C. 831

61. Art; 54th Monarch c.811 BC, slain by his successor, uncle of the 53rd Monarch

62. Olioll Fionn

63. Eochaidh

64. Lughaidh Lagha; died 730

65. Reacht Righ-dearg; 65th Monarch, c.653 BC, so called the Red King due to his
having a hand in the slaying of Queen Macha of the line of Ir. The only woman
who was a Monarch of Ireland. He subdued the Pictish nation in Scotland. Died B.C. 633.

66. Cobthach Caomh

67. Moghcorb

68. Fearcorb

69. Adhamhra Foltcain; died B.C. 412

70. Niadhsedhaman; 83rd Monarch cc.319 BC. In his time, through "the sorcery and
witchcraft of his mother, the wild deer were usually driven home with the
cows and tamely suffered themselves to be milked every day".

71. Ionadmaor; 87th Monarch c.218

72. Lughaidh Luaighne; 89th Monarch cc.198 BC

73. Cairbre Lusgleathan

74. Duach Dalladh Deadha; 91st Monarch c.168 BC

75. Eochaidh Garbh

76. Muireadach Muchna

77. Mofebhis; his wife. (A mistake here that O'Clery decided to leave as is.
She was entered in the Irish Regal Roll instead of her son, Loich, and
O'Clery did not choose to disrupt the sequence of numbers.)
78. Loich Mor

79. Eanna Muncain

80. Dearg Theine; He had a competitor, Darin, in the Kingdom of Munster, of
the line of Ithe. Ithe was the uncle of Milesius and the first
(Milesian) discoverer of Ireland. They took turns being Monarch with the other one
being governor of civil affairs.

81. Dearg

82. Magha Neid

83. Eoghan Mor [Owen Mor] or, Eugene The Great. A wise prince and great
warrior. He battled continually with "Conn of The Hundred Battles", the
110th Monarch in A.D. 122. Finally they divided the Kingdom into equal
parts. He was eventually slain by Conn.

84. Olioll Olum. His second son, Cormac Cas, branches off to BRIAN BORU.
Olioll Married the daughter of Conn, who had slain his father. She was a
widow of a chief of Conn's territory and her son demanded of Olioll that
he should benefit from the agreement of their ancestors. Olioll refused
and banished Maccon out of Ireland. He retired to Scotland and there
soon collected a strong party of friends and relations. With the help of his
Ireland relations he made war upon Olioll. The Monarch Art-Ean-Fhear's
forces joined Olioll in the great and memorable battle against Maccon at
Magh Mucromha, near Athenry, where Art and seven of Olioll's nine sons,
by Sabina, died. Their army was totally defeated. By this victory, Maccon
recovered his right to the Kingdom of Munster, and became Monarch for 30
years, leaving the Kingdom of Munster to his stepfather Olioll Olum,
undisturbed. Olioll had two sons left, Cormac Cas and Cian. Olioll
learned that after the death of his son Owen Mor, a son had been born to
him named Feach. From Cormac Cas came the O'Briens, MacMahons,
O'Kennedys and other nobility of Thomond. From Owen Mor came M'Carthy, O'Sullivan,
O'Keeffe, and the nobility of Desmond. From Cian came O'Carroll,
O'Meagher, O'Hara, O'Gara, etc.

85. Owen Mor

86. Fiacha (or Feach) Maolleathan

87. Olioll Flann-beag; King of Munster for 30 years

88. Lughaidh

89. Corc; from him the city Cork was named. To shun the unnatural love of his stepmother, he fled in his youth to Scotland where he married the
daughter of the King of the Picts. One of his several sons was Main Leamhna who remained in Scotland and was ancestor of "Mor Mhoar Leamhna" i.e., Great Stewards of Lennox; from whom descended the Kings of Scotland and England of the Stewart or Stuart Dynasty.

90. Nathfraoch

91. Aongus or Aeneas; first Christian King of Munster, had 24 sons and 24 daughters. This King was baptised by St. Patrick. Offering to plant his Staff, or Crozier in the ground, the Saint accidently pierced the foot of Aeneas, whereby he lost much blood; but thinking this was a part of theceremony, he patiently endured it until the Saint was done. (But this same story is also told:
"Eochaidh, son of Fiachra, son of Eoghan, was baptised with Eoghan; during the ceremony the Apostle's Staff is said to have accidently pierced thenaked foot of of the prince.")

92. Felim; second Christian King of Munster

93. Crimthann 125th Monarch A.D. 365

94. Aodh Dubh [Duff]; reigned 15 years

95. Failbhe Flann. Reigned 40 years. He had a brother Fingin who reigned before him and there is dispute as who was the eldest. Because of this the Heber line ends here and splits off to MacCarthy from Failbhe, andO'Sullivan from Fingin.