The name might seem odd but it refers only to the drying of organic solution. What it refers to is the removal of trace amounts of water from organic solution.
You can find more information on drying solutions here.
Many organic solvents are immiscible with aqueous solutions, but they are able to dissolve significant amounts of water because of their polarity i.e., diethyl ether dissolves 7 % of its mass in water!
In most cases this is of no importance but some procedures are very sensitive to the presence of seven small amounts of water.
Generally, the more polar the solvent is, the more hygroscopic it will usually be because it dissolves the water better.
The document linked to discusses various ways of drying solutions.
