The Washington District tartan was adopted as the official state tartan of Washington in 1991. A project of the Vancouver Country Dancers, it was designed by Margaret McLeod van Nus and Frank Cannonito to commemorate the Washington State Centennial celebrations. The tartan is accredited by the Scottish Tartans Society.

The background color of green represents the rich forests of Washington ("The Evergreen State"). The perpendicular bands of contrasting colors are symbols for the following features of Washington state:

blue for the lakes, rivers and ocean;
white for the snow-capped mountains;
red for the apple and cherry crops;
yellow for the wheat and grain crops
black for the eruption of Mount St. Helens

Tartan refers to fabric woven into plaid patterns representing clans (families) or regions in Scotland.

In old Scotland, the tartan was used for clothing and as a banner or flag. Because a family or community worked the cloth together, their clothing was made of the same patterns, and so a person could be recognized by the Tartan plaid that he wore. Tartans can also be designed as a symbol commemorating a special event or person.

Source:

State Symbols of Washington: Washington State Legislature
Tartans of Scotland
Scottish Tartans Museum; History

Links:

Cultural Symbols - 50 states