In general, the term washing means to remove something. For example, you have a water insoluble solvent containing some ionic contaminants, and you wash this with water to remove those water soluble ionic compounds.
As the first comment suggests, washing with a strong base like $\ce{NaOH}$ suggests that you are trying to remove an acid. A strong base would remove any acid, strong or weak. The base would be in an aqueous layer, and so there would also be the removal of water-soluble compounds.
The key is that your solvent must be non-miscible in water! In order to wash, your solvent and water cannot mix, so that the water pulls out the water soluble compounds and purifies the non-water soluble solvent.