From what I've seen, most rooms/shares in the area range from $450-$750, with most being around $550 or $600, including utilities and without pets. The area directly around the UW is called the UDistrict, and it's going to be a little expensive because of the tens of thousands of undergraduates vying for housing (not an exaggeration, the UW is huge). If you do find something at a low price in the UDistrict, it's going to be a terrible rat-infested cellblock in the basement, pretty much guaranteed.
The nearest residential neighborhood is called Wallingford, which is going to be cheaper and much nicer, and it's the neighborhood most grad students try to live in. It'd be possible to find a 1Bd apartment there for $1100, but I wouldn't bank on it... You're best bet is to look for a 2+ bedroom apartment - most 2 bedrooms end up going for about 600 or more a room, and most 3 bedroom apartments go for 400 or 500 a room, and those are pretty typical prices for all over Seattle.
Right near Wallingford is Fremont, which is a little more upscale with a bunch of quirky shops and restarants, plus a chocolate factory (!). The rent is a little higher there, you're probably better off living NEAR Fremont than in Fremont.
The further north you go on the West side of Greenlake (Ballard, Phinney Ridge, Aurora), the cheaper it gets, but you have to make sure you either have a car or are near a bus stop because it quickly becomes a pretty long walk to campus. However there's a lot going on in those neighborhoods and they all seem pretty cool to live in.
On the east side of Greenlake there's Ravenna, which can be cheap but I don't know anything about it, and typically I think the neighborhoods get "nicer" the further north you go, and it gets pricier. Northgate I mostly know because of the big mall up there, but it seems kind of... way far North. But compared to amoryb's commute from Olympia, it's probably not TOO bad. Either way I have no idea about the quality of living or prices up there.
Also to the South there's Capitol Hill, where I live now. The prices are about the same as Wallingford but it's a little further from campus. There's a lot of unique neighborhoods in Seattle, and capitol hill is definitely the most, uh, loud and proud, if you know what I mean? Capitol Hill dwellers are definitely a unique brand of freaks and geeks.
You can find cheaper and cheaper housing the further south you go, but I wouldn't recommend it. Particularly, if you're looking up housing on craiglist and the neighborhood is Yesler Terrace or the International District, you probably wanna look elsewhere. They tend to be poorer neighborhoods with higher crime rates, and all the other unfortunate realities of inner city life. But the ID has killer pho, if you're into that.
Central Seattle, Queen Anne, and Downtown are all really expensive and way out of your way, don't bother looking there.