A boarder is a person who rents a room, either from a private family or a boarding house. The price of the room usually includes meals, thus you are paying for "room and board". The term board, meaning meals, comes from medieval times when the great hall of the castle was turned into a dining room for the duration of a meal by trestle tables, which basically amounted to boards laid on sawhorses. At that time, many people worked in exchange for "clothing, room, and board", without receiving actual money for their labor.
Remember the difference by knowing that boarder and meal both have an "a" in them, and border and country both have no "a".