This page (originally here, now available only from the web archive) may help:
[soft state] is information (state) the user put into the system that
will go away if the user doesn't maintain it. Stated another way, the
information will expire unless it is refreshed.
By contrast, the position of a typical simple light-switch is
"hard-state". If you flip it up, it will stay up, possibly forever. It
will only change back to down when you (or some other user) explicitly
comes back to manipulate it.
The BASE acronym is a bit contrived, and most NoSQL stores don't actually require data to be refreshed in this way. There's another explanation suggesting that soft-state means that the system will change state without user intervention due to eventual consistency (but then the soft-state part of the acronym is redundant).
There are some specific usages where state must indeed be refreshed by the user; for example, in the Cassandra NoSQL database, one can give all rows a time-to-live to make them completely soft-state (they will expire unless refreshed), but this is an unusual mode of usage (a transient cache, essentially).
"Soft-state" might also apply to the gossip protocol within Cassandra; a new node can determine the state of the cluster from the gossip messages it receives, and this cluster state must be constantly refreshed to detect unresponsive nodes.