Little is known about them except legends. Still those agree on some points.
Desolations came in cycles, they were regular in coming. During Desolations, the inhabitants of Roshar were attacked by an enemy force, generally called Voidbringers. The cycle of Desolations seemed somehow attached to the Oathpact.
One certain Desolation lasted eleven years.
Actually, at the end of what was called the Last Desolation, nine out of ten Heralds abandoned the Oathpact leaving only Taln to return to damnation, the place where the Heralds went between the Desolations. This led to a very long 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.