There are a mind numbing number of Chiang Mai guesthouses and apartments renting by the day week or month. Estimates are around 150 with about 3,000 rooms available. At the lower end of this group ($) you can expect a fairly bare room with a bed and fan and the bathroom down the hall renting for 100 Baht (about 3 U.S. Dollars) per night.
Towards the upper end ($$$), you should find a clean and comfortable, nicely decorated room with private hot water bath, TV, air-conditioning, fridge and phone. One of these in a good location, may rent for 600 to 800 Baht per night depending on the length of stay. Most will have a small restaurant attached with simple bar service, often in a garden setting.
Near the top of the range for Chiang Mai guesthouses there is usually little difference from small hotels.
Some recommendations:
Daret's Guest House
Across the street from the city moat just a few steps from the Thapae Gate plaza. Has a popular, inexpensive restaurant downstairs.