Little is known about them except legends. Still those agree on some points.
Desolations come in cycles, and are regular in coming. During Desolations, the inhabitants of Roshar are attacked by an enemy force, generally called Voidbringers. The cycle of Desolations seems somehow attached to the Oathpact.
One of the Desolations lasted eleven years.
At the end of what is called the Last Desolation, nine out of ten Heralds abandon the Oathpact leaving only Taln to return to damnation, the place where the Heralds go between the Desolations. This leads to a period of about 4,500 years without a Desolation.
Knights Radiant and Desolations
According to Vorinism, mankind fought the Voidbringers during Desolations ninety and nine times, before mankind finally won and drove them back to the Tranquiline Halls. That victory was later referred to as the Last Desolation.