For short stays the faculty has an agreement with the Hotel Mora, with a price of about 50 euros per night.
The hostels are generally cheaper than hotels, and there are a lot of them in Madrid. Some of them have rooms with WC and shower, some do not. Most of them have a reception, and nowadays many of them come with free wireless access as well.
Below a list of hostels where we have had guests staying, and from which they have not left prematurely :-)
There are a number of web pages dedicated to listing Madrid hostels; one of our visitors has had good luck with the www.hostelworld.com web page.
The faculty has two (apparently infrequently updated) web pages with hostels and flats as well, but the Babel group has no own experience regarding these places.