The washing of hands is required before eating a meal at which bread is served. One reason for hand washing is purely mundane: to clean them before eating. The origin of this law, however, is biblical (Exodus 30:17-21). Moses was commanded to make a copper laver and to place it at the entrance to the alter area so that Aaron and his sons could wash their hands before approaching the alter to offer sacrifices. Verse 20 concludes, ". . .and it shall be for them a statute forever." The washing of hands makes them ritually pure. This symbolic expression of washing away impurity from one's hands dates back to Temple times, when the Priests (Kohanim), as described above with Aaron, devoted their lives to the Temple and its sacrificial system. Before performing any ritual, a Kohein was required to wash his hands, making himself pure and ready to offer a sacrifice.
When the Temple was destroyed in 70 CE, the table in a home came to represent the Temple alter. The bread placed on it symbolizes the offerings that were once brought to the Priests. The Sages, who believed that the Temple and the Priesthood would be restored, did not want the practice of washing hands before handling an offering or performing a ritual to be forgotten, and so washing the hands before a meal was strictly enforced. Today, Orthodox and traditional Jews still wash their hands before eating bread.
Sources: Kolatch, Alfred J. The Second Jewish Book of Why. Jonathan David Publishers, Inc.; Middle Village, New York, 1985.