To launder is to wash your clothes or other things made from cloth, like sheets and towels. A college student might try to wait to launder her clothes until she goes home for Thanksgiving.
When you launder, you wash a load of laundry, or dirty clothes. These days most of us use a washing machine to launder, adding detergent to the water and then tossing the wet things in the dryer after they're clean. Another meaning of launder is "hide the origins of illegal money," or "make dirty money look clean," which is usually done by making it appear that the money was earned through a legitimate business.