In that context, it means the removal of equipment, plugging of the well with cement, any environmental clean-up, etc.
The objective of well abandonment is to ensure that no hydrocarbons leak into surface water or into the atmosphere.
Wells that have been used as injectors or have been subject to fracking operation are more likely to develop leaks because the injected substances can create channels that permit uncontrolled flow outside the casing.
The term's application has been broadened from its original context to apply to the abandonment of other business ventures, primarily in manufacturing.
For example, General Motors had some abandonment costs from shutting down the Pontiac and Saturn brands.