There is plenty of off-campus housing nearby in the University District. For those choosing to live farther away from campus, it is easy to commute by bike or Seattle's Metro bus service.
Seattle is different than some U.S. cities in that the central neighborhoods here are still considered very desirable places to live and tend to be more expensive than the more distant and / or suburban locations. The neighborhoods of Capitol Hill, Wallingford, Green Lake, Ravenna, and Fremont are all great places to live and are a short bus or bike ride from the University. The farther from the city center you go, the lower prices tend to be. Those looking for the least expensive apartments will find them in the north and south end suburbs but will have a longer commute and less interesting surroundings.
Advertisements for shared and single-occupancy housing are readily found posted at the University or in local papers. It can be very difficult to find a place during the weeks just prior to and after the beginning of Autumn quarter. Prices for shared housing in houses and apartments range from $300 to $600 per month, depending on the quality of housing and the distance from the University. Single-occupancy housing prices range higher, with studios starting around $550, and one bedrooms starting around $700 per month.
Here are some links to rental listings:
The Seattle Post-Intelligencer classifieds
UW off-campus housing listings
UW on-campus housing
The Stranger classified ads
Seattle Weekly classified ads