Budget
Jasmine International Youth Hostel
Clean place, very friendly staff and guests with decent levels of English; you might find only Chinese patrons here and most seem to come just because they like the hostel as opposed to the city. Free Wi-Fi on the ground floor and in the entertainment room. Bar and small menu if you want a quick bite without venturing outside. Bathrooms are all shared and towels are available for purchase. Centrally located, <5km to most sites in all directions close to Nanjing University and Grand Hotel.
Nanjing Danfeng International Hotel
Easily the best budget option in Nanjing. Private rooms are spotlessly clean and excellent size for the money. Although their listing on numerous websites says it is "wireless", each room is equipped with wired Internet connections you can borrow an Ethernet cable from the reception desk. Those without can share the communal computer for free. Don't share a private room with anyone you don't want to see naked, as the bathroom/toilet is housed inside a strange transparent glass enclosure.
Xiyuan: Nanjing University Foreign Student Dorm
Located in a central part of town, there are plenty of shops, bars, and restaurants in the area. The rooms in this 15-floor building are dorm style: 2-3 share a room, while a common bathroom squat toilets and small kitchen is shared by the entire floor. Beginning with floor 11 the rooms got their own small bathrooms with toilet and shower. A kitchen-like facility though is available one the floor.