Evaporite mining usually refers to a method of extracting salt minerals from brine (sometimes called saline) solutions. These solutions may come from solution mining or collection of solutions from particularly concentrated saline waters.
Essentially, the solution is pumped into evaporation ponds that have characteristically high surface area to volume (i.e., large ponds that are very shallow). As water is evaporated from the pond, the salt minerals become concentrated and can then be removed. Clearly, these types of operations work best in warm, dry climates with relatively large, flat areas available for pond construction.
It should also be noted that "evaporite mining" can sometimes refer to other mining processes used to mine evaporite deposits (i.e., sedimentary deposits that were formed when an old seabed or collection of seawater dried).