Much of their early history is unclear, but it is believed that many thousands of years before their invasion of the Star Wars galaxy, the Vong were a peaceful race that lived in symbiosis with their homeworld Yuuzhan'tar, which itself was a living entity.
Invaded by a race that was more machine that organic, Yuuzhan'tar showed the Vong how to defend themselves, and provided bio-technological tools to do so. While they were able to do so this had the unfortunate effect of awakening their war like tendencies.
The Vong went on to enslave or destroy many of the species in the home galaxy before turning on each other. One of the casualties of their civil war was Yuuzhan'tar - which was destroyed in the war.
Before that was done Yuuzhan'tar dispatched a seed towards the Star Wars galaxy in the hopes that it would take root and become another living world.
Stripped of their symbiosis and their connection to the Force, the Vong experienced intolerable pain. Pain to them became the only way to experience what they had lost, and became central to Yuuzhan Vong society.
The Vong were soon forced to leave their home galaxy and journey for greener pastures, a journey that would take several thousand years. As early as about 4,000 years before the Battle of Yavin Vong scouts reached the galaxy.
Revan's colleague Canderous Ordo once referred to encounter he had with this scout. During the long journey the worlds fleet nearly destroyed itself with the domains of the Vong fighting against each other, and was in bad shape by the time of the invasion.
By the second decade after Yavin, the main force reached the galaxy, and were determined to cleanse the galaxy of all machine technology, conquer all living species in the galaxy, and exterminate the Jedi.
When Supreme Overlord Quoreal began having second thoughts about invading the galaxy when scouts reported the existence of a living world, a coup lead by Shimrra Jamaane toppled Quoreal's rule and allowed the invasion to proceed. With the Vong unaffected by most Force powers, the Jedi were suddenly uncertain how to face this new enemy. Worse, many in the galaxy turned against them - and groups such as the Peace Brigade tried their best to help the Vong rid the galaxy of the Jedi.
Over the next five years the Yuuzhan Vong War proved to be one of the most destructive conflicts ever visited on the denizens of the Star Wars galaxy. Over 365,000,000,000,000 sentient beings died in the war. Countless worlds were ravaged, most notably Ithor, a lush jungle paradise planet with inhabitants that vowed to keep unspoiled. The Vong unleashed toxic chemicals on Ithor, and in less then five minuts, Ithor was uninhabitable.
Even Coruscant itself fell to the Vong for a period of time during which Shimrra took up residence on the planet, which the Vong tried to reshape into a copy of their original homeworld. Many were enslaved and forced by surge coral to go fight for the Vong.
When a Jedi strike team was finally able to kill Shimrra - who turned out to be a puppet of the Shamed One Onimi, and Onimi killed a short time later by Jacen Solo a large number of Vong committed suicide. The remaining Vong realized that continued fighting would be useless, and surrendered themselves to the Galactic Federation of Free Alliances.
Post War Era
The Yuuzhan Vong were allowed to settle on Zonoma Sekot - the seed that the original Yuuzhan'tar had dispatched all those years ago. Sekot then left known space for a star system where the Vong would be able to rebuild their society, safe from reprisals from some in the galaxy who wanted to take revenge on and exterminate the Yuuzhan Vong.
About 100 years later the New Jedi Order became involved in an effort to improve relations between the Vong and the citizens of the galaxy. Using the Jedi world of Ossus as a test bed, the Jedi allowed the Vong to reshape part of the world to see if the Vong could help rehabilitate devastated worlds. When the results were showed off to the galaxy worlds vied for the chance to allow the Vong to rebuild their worlds as well.
Unfortunately Darth Krayt and his Sith order sabotaged the project, causing people on worlds being reshaped to grow bony growths and plants became deformed. Millions died as a result of the Sith sabotage. The galaxy turned against the Vong again, and the Jedi who supported them as well. This set the stage for the Empire to declare war on the Galactic Federation of Free Alliances, and defeated them after a brief conflict. The Sith then turned on the Empire, and took control of the galaxy once again.