what does this left hand represent?
The high kings.
Coats of arms
and it is because of this prominence that the red hand (though a right hand is used today, rather than the left used by the high kings) 
the wolf and the lion game of thrones episode Fifth episode can you mention any characters from this episode? Lord Eddard Stark, Tyrion Lannister, Catelyn Stark can you mention any actor from this episode? Finn Jones, Kate Dickie, Lino Facioli, Brendan McCormack were some of them. where was the episode shot? The Paint Hall studio, Shane's Castle, Fort St Angelo (Maltese town of Vittoriosa). the castle that you just mentioned, is it a real castle? Yes, it is a ruined castle near Randalstown in County Antrim, Northern Ireland. when was it built? 1345 by whom? A member of the Clandeboy O'Neill dynasty. has it served as the filming location for any other works? UNANSWERABLE what are the origins of the dynasty that you just mentioned? Irish Gaelic origin were they rulers? Yes, they were Kings of Cenél nEógain. do they have any successors in foreign nations? Yes, in Scotland. what is their family crest? The coat of arms were white with a red left hand  (though a right hand is used today). The red hand by itself has become a symbol of the O'Neill name.
A promise of land to the first man to sail or swim across the sea and touch the shores of Ireland. The red hand by itself has become a symbol of the O'Neill name, such that when other O'Neill family branches were granted or assumed a heraldic achievement, this red hand was often incorporated into the new coat of arms in some way. The red hand representa O'Neill left hand, that he cuts and throws it onto the beach before the other challengers of a race can reach the shore. This entitles him to become the first to touch land and win all of Ireland as his prize.
O'Neill dynasty O'Neill dynasty O'Neill dynasty
Coats of arms Coats of arms Coats of arms

The red hand is explained by several legends, with a common theme but of a promise of land to the first man to sail or swim across the sea and touch the shores of Ireland. Many contenders arrive, including a man named O'Neill, who begins to fall behind the others. The red hand by itself has become a symbol of the O'Neill name, such that when other O'Neill family branches were granted or assumed a heraldic achievement, this red hand was often incorporated into the new coat of arms in some way. The red hand is explained by several legends, with a common theme but of a promise of land to the first man to sail or swim across the sea and touch the shores of Ireland. Many contenders arrive, including a man named O'Neill, who begins to fall behind the others. O'Neill cuts off his left hand and throws it onto the beach before the other challengers can reach the shore, becoming the first to touch land and win all of Ireland as his prize. These legends seem to originate (or to have been written down) in the 17th century, centuries after the red hand device was first used by O'Neill families.
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O'Neill dynasty [SEP] Coats of arms
It is a common misconception that there is one coat of arms associated to everyone of a common surname, when, in fact, a coat of arms is property passed through direct lineage. This means that there are numerous families of O'Neill under various spellings that are related, but because they are not the direct descendants of an O'Neill that owned an armorial device do not have rights or claims to any arms themselves. The coat of arms of the O'Neills of Ulster, the branch that held the title of High Kings of Ireland, were white with a red left hand (latterly, the Red Hand of Ulster), and it is because of this prominence that the red hand (though a right hand is used today, rather than the left used by the high kings) has also become a symbol of Ireland, Ulster, Tyrone and other places associated with the family of O'Neills.