Adding up to Justin's answer (After asking him to do it himself and waiting) to provide historical and political context which might be needed to understand the question and the answer, here goes.
Volantenese Political Structure
For all intents and purposes, Volantis is closest to ancient Valyria in term of administration. It is a freehold where a number of families rule together. In other words, we can say it is like merchant republics in our world.
There are two main factions of those families:
Tigers: The tigers originate from those of the Old Blood who favored conquest at the start of the Century of Blood. Some years the tigers elect a triarch, other years there is no member of the tiger party elected triarch. The tiger cloaks, the Volantene guards, are presumably named after the militant tiger party. The tiger party has not held power for the last three hundred years, ever since their fall in aftermath of century of blood. They are supported by families of old blood of Valyria.
Elephants: The elephants are the party of the merchants and moneylenders, advocating trade. Some of the first elephants were women. The elephant party has held power for the last three hundred years, during which at least two of the three triarchs in power have been elephants, due to the general Volantis resentment against Tigers for waging the war in century of blood.
Historical Context - Century of Blood
After doom of Valyria, began the era called "Century of Blood".
Volantis, under rule of the Tigers, considered herself to be heir to legacy of Valyria and went to war to conquer all Valyrian territories.
They conquered Lys and Myr into their "New Valyria". They also sent expeditions to recover lost Isles of Valyria.
Afterwards they marched to next target, Tyrosh. That's when Pentos realized that if Tyrosh fell, they would be next. So they joined Tyrosh. While Volantis was busy holding off Tyrosh and Pentos, Lys and Myr rose up in rebellion to win their freedom.
Braavos, Qohor, Norvos, Storm King Argillac the arrogant1 and Young Lord Aegon Targaryen on his dragon Balerion also joined the fray against Volantis as creation of a New Valyria threatened all of them and their interests. Only Lorath remained neutral, mostly because they were on the Farthest edge of the conflict and have historically remained isolated. Following were their reasons I suppose, also explained in another answer:
Sealord feared that a new Valyria would try to enslave Braavos because Braavosi were descended from run-away slaves of the Valyrian Freehold.
The Nine Free Cities are the daughters of Valyria that was, but
Braavos is the bastard child who ran away from home.
Qohor and Norvos were naturally concerned about their freedom and unwilling to accept Volantenese supremacy.
Aegon Targaryen was concerned about his own holdings as Dragonstone was also a part of old Valyria. Also on account of being one of the old blood and Dragonriders, he probably considered the claims of Volantis to be heir of Valyria as ridiculous. But since Aegon never tried to rebuild Valyria and instead turned to forge his own new realm, we can say he never had any such ambition.
Storm King felt threatened as in case of a successful reunification, Volantis would be a direct threat to his Kingdom, right across the narrow sea. Who was there to ensure that New Valyrian conquests won't spill across the narrow sea, well beyond the original boundaries of the freehold?
Lorath did not feel inclined to join the fight because they were an Island nation, cut off from most of the continent Essos. Not to mention they always maintained low profile in regional affairs. However, ordinary Lorathi citizens did fight on side of Braavosi or Norvosi despite neutrality of the government.
End was like German Revolution of 1918, peace favoring Volantenese faction Elephants wrested the power away from War-loving tigers, and ended their ambition to reunite Valyrian territories.
Since the war cost Volantis a lot in terms of human and material resources, the followers of Elephants went on a rampage and struck heads of the statues of those historical tiger figures whom they thought responsible for all the death and misery they suffered.
This is why many statues are headless in Volantis. This is mentioned in A Dance With Dragons, Chapter 22, Tyrion VI.
1. Funnily enough, this was the war when Argillac the Defiant had a chance to know Aegon Targaryen. He was so impressed by the prowess of the scion of Valyria, he planned to forge an alliance with Aegon against the Ironborn and to seal it, offered the hand of his maiden daughter and heiress. Aegon however, being already married, suggested that the Storm Princess and Orys Baratheon should marry. Argillac was enraged at the insult of being offered hand of a bastard for his daughter and cut the hands of Aegon's envoy, thereby instigating the Targaryen conquest of Westeros and end of his House in male line.