The BASE acronym was defined by Eric Brewer, who is also known for formulating the CAP theorem.
The CAP theorem states that a distributed computer system cannot guarantee all of the following three properties at the same time:
Consistency
Availability
Partition tolerance
A BASE system gives up on consistency.
Basically available indicates that the system does guarantee availability, in terms of the CAP theorem.
Soft state indicates that the state of the system may change over time, even without input. This is because of the eventual consistency model.
Eventual consistency indicates that the system will become consistent over time, given that the system doesn't receive input during that time.
Brewer does admit that the acronym is contrived:
I came up with [the BASE] acronym with my students in their office earlier that year. I agree it is contrived a bit, but so is "ACID" -- much more than people realize, so we figured it was good enough.