Mid-priced accommodation
Homely and comfortable
Aside from the ubiquitous chain hotels, which York has both near the city centre and out of town, the standard mid-range accommodation is provided by Bed and Breakfsts. These tend to mean around half a dozen rooms in the host's house. Gueshouses are very similar, usually just a few rooms bigger.
(North American visitors should be aware that these are more modest offerings than bed & breakfast offerings stateside.)
Breakfast almost always means a full fooked breakfast, if you want it, as well as cereals and toast, all of which should set you up for a good part of the day.
If you're drawing a blank, don't want breakfast included, then consider a homestay via a site like Airbnb.
For general booking information see Finding Accommodation.
Guide prices for a double room from around £50 to around £120. Chains such as Travelodge or Premier Inn charge around £70 - £90.