This is a term mostly found in marine insurance, where it refers to the act of
surrendering the subject matter insured to the insurers in return for a total loss
payment in certain circumstances. This is quite standard in marine practice, but in
other classes of property insurance, policies usually specifically exclude
abandonment.
The important thing to be remembered with abandonment is that the subject
matter insured (or what is left of it) is completely handed over to the insurer, who
may therefore benefit from its residual value.