Seattle's central neighborhoods are considered very desirable places to live but tend to be more expensive than more distant 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. Ballard and Queen Anne are a couple of other very nice neighborhoods you might consider, although they are slightly farther from the university. Lake City and Greenwood have lots of apartments and are somewhat less expensive than the other neighborhoods mentioned. They are also a bit farther from UW and less charming. Northgate is not known for its charm, but it does have decent bus service to UW and has lots of apartments. There is plenty of off-campus housing nearby in the University District. It's very convenient, but housing tends to be a bit rundown. There are new on-campus graduate housing options for those who prefer to go that route.
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 (not the eastside suburbs - due to Microsoft et al, housing in Bellevue, Kirkland, and Redmond is as expensive or more than central Seattle) but will have a longer commute and less interesting surroundings. If you do opt for less expensive housing in the suburbs, you'll want to choose one that has frequent all-day bus service to the UW - the north King County suburbs of Kenmore and Bothell fit this requirement (all day bus service on route #372 along highway 522) and Bothell in particular has reasonable amenities.
Prices vary, but in Seattle one bedrooms start around $1300 per month and two bedrooms start around $1600...and go much higher. Seattle has become expensive in recent years, but remind yourself it could be worse - this isn't the bay area.