In the context of multiuser computer systems (like shared mail servers in universities), soft quota is the amount of storage space that is considered reasonable for a user. The user is allowed to exceed his soft quota for a grace period, during which he gets warning messages.
Once the grace period expires, various (unpleasant) things can happen. One is that the hard quota gets set to the soft quota value, effectively rendering the user unable to write anything to his storage.
See also: hard quota, quota, quota policy and UNIX.