Put simply, a Calamity is any sufficiently disastrous event on our world. When that happens, Hydaelyn desperately needs aether to heal the damage and maintain balance. The only readily available source of sufficient aether are the parallel Shards--Hydaelyn is forced to reabsorb the Shard back into our world. The Ascians plan to force Hydaelyn to reabsorb all the Shards, and so prepare the Source for the reawakening of Zodiark. The seven past Calamities have thus seen Hydaelyn rejoin seven Shards to the Source.