During the Desolations, the Odium had given his soldiers powers, allowing them to remain past their deaths as Cognitive Shadows. After being killed, these singers, known as the Fused, were able to take over the bodies of other singers and manipulate the Surges with Voidlight. Everytime a Fused was killed, its soul it would simply be born again into a new host. This formed an impossible challenge for the humans as their strongest enemies could never be permanently killed, and so came the solution: the Oathpact.
Five men and five women gave themselves up voluntarily to this pact. With Honor's blessing, they traveled to Braize and sealed the spren of the enemy there. It prevented the Fused from returning, and so ended the war. Or so Honor thought. While gods and spren cannot break oaths, men can and will. The Fused captured the Heralds and tortured them. Eventually, one of the Heralds would break, starting a new Desolation. At the beginning, the Heralds were strong and hundreds of years passed between Desolations, but near the end, their souls wore thin. There were only 2 years between the last two Desolations. After the Aharietiam, the Heralds realized that after all that time, there was one Herald that had never broken the oath and started a Desolation. Talenelat, the Bearer of Agonies. The other Heralds abandoned the Oathpact, leaving only Talenelat to uphold it by himself. For over four and a half millennia, Talenelat withstood the torture of the enemy, but he finally broke, joining the other nine who still live. This caused the True Desolation, currently occurring on Roshar.
After the Last Desolation, nine of the Heralds thought they abandoned the Oathpact, leaving only Talenel'Elin to uphold it. However, despite what they believe, all the Heralds are still bound to the Oathpact.
The Heralds and the Shard Honor were the only two parties who were part of the Oathpact. Odium being bound to the Roshar System was not a direct result of the Oathpact but the Oathpact is part of it. 
While at least some of the Oathpact still stands, it is unknown exactly how much remains.