a song of ice and fire - What is the 'Dance of the Dragons'?

There is an event in A Song of Ice and Fire referred to as the 'Dance of the Dragons'; what was this event and what does it mean? and what was the final outcome?

^{Note: Not to be confused with the book A Dance With Dragons.}

Answer

The Dance of the Dragons was the second Targaryen civil war which occurred between the years 129 and 131 AC. The event is also known as "The Greens vs The Blacks".

The war was sparked by inheritance complications between Rhaenyra Targaryen and Aegon II Targaryen after the passing of their father King Viserys I Targaryen.

Cause
Rhaenyra was the eldest child (daughter) of Viserys I through his first marriage to Aemma Arryn. She was also groomed and designated as the Heir Apparent by Viserys himself.

Aegon II was the eldest son of Viserys I, but through his second marriage to Alicent Hightower.

Outcome
In the end, both claimants (as well as their respective dragons) were killed.

After this event, the Targaryens followed the Male-preference cognatic primogeniture rules of inheritance, which means that only males have the preference of inheritance - females can only inherit if their is absolutely no males left in the line.

Note: the rest of Westeros does not follow this style of inheritance; see my break-down of inheritance.